Design and control methods for high-accuracy
variable reluctance actuators
by
ARCHNES
AMASSACHUSETNNSTTUTE
JUL 3 0 2015
IBI
LIBRARIES
Ian MacKenzie
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2015
June 2015
@ Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
Author ...................................................
Department of Mechanical Engineering
February 13, 2015
Signature redacted
Certified by..........................
David L. Trumper
Professor of Mechanical Engineering
Thesis Supervisor
Signature redacted
Accepted by...............................................
David E. Hardt
Chairman, Department Committee on Graduate Theses
I
I
I
I
________
_________
Design and control methods for high-accuracy variable
reluctance actuators
by
Ian MacKenzie
Submitted to the Department of Mechanical Engineering
on February 13, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
This thesis presents the design and control techniques of a variable reluctance actuator for driving a reticle motion stage in photolithography scanners. The primary
thesis contributions include the design and experimental demonstration of a magnetic flux controller that uses a sense coil measurement, the design and experimental
demonstration of a novel method to estimate actuator hysteresis in real-time, and the
development of an actuator model that incorporates the effects of eddy currents.
The reticle stage in a scanning lithography machine requires high accelerations
combined with sub-nanometer position accuracy. Reluctance actuators are capable
of providing high force densities (force per moving mass) and lower power values
relative to the present state-of-the-art Lorentz actuators that are used to drive the
reticle stage. However, reluctance actuators are highly nonlinear with both current
and air gap. They also display other nonlinear behavior from hysteresis and eddy
currents. Linearizing the reluctance actuator is required for the high force accuracy
required in the scanning stage.
In this thesis, we present a way to linearize the reluctance actuator with flux
control using a sense coil as the feedback measurement. Because the sense coil is
AC-coupled, we design a low-frequency estimate of the magnetic flux based upon the
actuator current and air gap measurements. We combine the low-frequency estimate
with the sense coil measurement using a complementary filter pair that provides an
estimate of the flux from DC to frequencies of several kHz. For the low-frequency
estimate, we develop a novel method for estimating the actuator hysteresis in realtime. For this flux estimator, we use an observer to model the actuator flux which
treats the changing air gap as a disturbance to the plant model. The use of an
observer allows the identification of a single-variable hysteresis model of actuator
current rather than a two-variable hysteresis model of current and air gap. We also
introduce a novel way for expressing the actuator hysteresis, whereby we incorporate
the linearizing effect of the air gap directly into a Preisach hysteresis model via a
change of variables. We demonstrate experimentally that this method is numerically
stable in the presence of a dynamically changing gap, in contrast to some alternative
3
methods.
We designed and built a reluctance actuator prototype and 1-DoF motion testbed
to demonstrate the accuracy of the actuator models and control techniques. We experimentally demonstrated that we can achieve a flux control bandwidth of 4 kHz that
is capable of reducing the stiffness of the reluctance actuator to less than 0.012 N/iim
for frequencies up to 100 Hz. This results in a force error of less than 0.03% of the
full-scale force for a 10 pim air gap disturbance at this frequency. We also demonstrate
that the actuator hysteresis model is capable of estimating the actuator flux accurately in the presence of dynamic gap disturbances of at least 35 1m peak-to-peak
and with a static offset from the nominal air gap of at least 50 pm.
Thesis Supervisor: David L. Trumper
Title: Professor of Mechanical Engineering
4
Acknowledgments
I would first and foremost like to thank my advisor David Trumper for his guidance,
encouragement, and patience throughout my graduate years. I have learned a great
deal from him and have very much enjoyed working with him. He has imparted to
me not only his extensive knowledge of mechatronics, but also how to communicate
more effectively and how best to situate my research within the big picture.
I would also like to thank the members of my thesis committee. All of them offered
invaluable feedback and support. Professor Jeffrey Lang took a keen interest in my
project and helped me to debug experimental problems and to clarify my thinking. I
learned electromagnetics from Professor Markus Zahn, without which knowledge my
thesis would have been impossible. He also offered much-appreciated encouragement.
Professor Martin Culpepper provided advice for improving my experimental testbed,
and helped me to broaden my thinking by always encouraging me to think of other
applications for my research. All of my committee members were always willing to
be available and were eager to see me succeed.
In addition to my committee members, I would like to thank Professor Noel
Perkins, my Master's advisor at the University of Michigan. He stimulated my interest
in research and encouraged me to apply to MIT.
I would like to thank ASML for their collaboration and financial support. Steve
Roux first introduced me to this project during a summer internship and enthusiastically supported its continuation. I would also like to thank Chris Ward and Mark
Schuster, both of whom offered valuable feedback. I thank Charlie Griffin and Mark
Hill for their help in setting up my experimental testbed at ASML. I also thank ASML
for the employment opportunity they gave me in the middle of my PhD.
I also extend my thanks to Fred Sommerhalter, who built the reluctance actuator
prototypes for my thesis. He also worked with us in finalizing the prototype design,
where his motor expertise proved invaluable.
My labmates have helped make my time at MIT gratifying. Darya Amin-Shahidi
collaborated with me on parts of my thesis and was an excellent sounding board for
5
discussing research ideas. Mohammad Nejad was always willing to offer his design
and manufacturing expertise and was never shy about correcting my English. The
Persian hegemony formed by Darya and Mohammad helped prevent the lab from descending into chaos. I also enjoyed collaborating with Roberto Melendez who helped
me machine parts and assisted me in preparing for my thesis defense.
Jun Young
Yoon was always willing to take the time to assist me in various endeavors, whether
it was setting up an experiment or printing my thesis for me in my absence.
The
ongoing 'feud' between Roberto and Jun Young provided endless free entertainment
to the lab. I enjoyed working with Aaron Gawlik on an earlier project and debating
whether Michigan hockey or Minnesota hockey was better (Michigan, of course). I
enjoyed frequent brainstorming sessions and philosophical discussions with Dean Ljubicic, even if on the latter he remains inexplicably unconvinced. I am also grateful
to his family for feeding a poor graduate student on numerous occasions. Kevin Miu
helped me navigate the idiosyncrasies of MIT, more often than not with a healthy
dose of sarcasm included. Dan Kluk helped persuade me to join Professor Trumper's
lab, and perhaps just as important, taught me all I know about craft beer.
I also
thank Lei Zhou, Minkyun Noh, Phillip Daniels, and Joe Church for their support. I
also thank Laura Zanganjori for her administrative help and her friendly demeanor.
I cannot fully express my gratitude to my parents, who in addition to humoring
me by smiling and nodding whenever I talk about my research, have freely given
me their love and support throughout my life and have taught me responsibility and
strong moral principles.
Growing up, my siblings often provided the opportunity
for applying these moral principles, despite frequent reluctance on my part to take
advantage of these opportunities. For that, I both thank and apologize to my siblings.
Finally, I thank God, who as Creator and Sustainer of the natural world has
provided the raw material for my scientific endeavors, and who as the Logos, the
Source of all order and rationality, has made it possible for me to discover nature's
truths. As C.S. Lewis wrote, "God never meant man to be a purely spiritual creature...
We may think this rather crude and unspiritual. God does not...
He invented it."
6
He likes matter.
Contents
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
1.2.1
Actuator Configuration . . . . . . . . . . . . . . . . . . . . . .
38
1.2.2
Control
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
1.3.1
Reluctance Actuator Design . . . . . . . . . . . . . . . . . . .
45
1.3.2
Reluctance Actuator Modeling . . . . . . . . . . . . . . . . . .
46
1.3.3
Reluctance Actuator and 1-DoF Testbed . . . . . . . . . . . .
47
1.3.4
Loop-Widening Investigation . . . . . . . . . . . . . . . . . . .
49
1.3.5
Experimental Results . . . . . . . . . . . . . . . . . . . . . . .
49
1.1
Background
1.2
P rior A rt
1.3
2
35
Introduction
53
Reluctance Actuator Design
2.1
Electromagnetic Fundamentals . . . . . . . . . . . . . . . . . . . . . .
54
. . . . . . . . . . . . .
54
. . . . . . . . . . . . . . .
60
2.1.1
Reluctance Actuator Electromagnetics
2.1.2
Lorentz Actuator Electromagnetics
2.1.3
Comparison between Reluctance Actuator and Lorentz Actuator 64
2.2
Core Materials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2.3
Actuator Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . .
69
2.3.1
Force-flux Relationship . . . . . . . . . . . . . . . . . . . . . .
69
2.3.2
Hysteresis and Saturation
. . . . . . . . . . . . . . . . . . . .
70
2.3.3
Eddy Currents
. . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.3.4
Gap Dependency . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.3.5
Fringing Flux . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
7
2.7
71
2.4.2
Flux-Biased Linearization with a Permanent Magnet
73
2.4.3
Flux-Biased Linearization with an Electromagnet
2.4.4
Operating Regime
2.4.5
Target Geometry
2.4.6
74
. . . . . . . . . . . . . . . .
. . . . . . .
76
. . . . . . . . . . . . . . . . .
. . . . . . .
78
Laminating the Stator Core and Target . . . . .
. . . . . . .
78
Linearization via Control . . . . . . . . . . . . . . . . .
. . . . . . .
82
2.5.1
Standard Feedforward and Feedback Control . .
. . . . . . .
82
2.5.2
Sensors for Feedback Control
. . . . . . . . . .
. . . . . . .
88
2.5.3
Advanced Control Techniques
. . . . . . . . . .
. . . . . . .
98
Alternate Actuator Configurations . . . . . . . . . . . .
. . . . . . .
102
2.6.1
Double-sided Actuator
. . . . . . . . . . . . . .
. . . . . . .
103
2.6.2
Multi-DoF Actuators . . . . . . . . . . . . . . .
. . . . . . .
105
2.6.3
Multi-pole Actuators . . . . . . . . . . . . . . .
. . . . . . .
107
. . . . . . .
108
.
.
.
.
.
.
.
.
.
. . . . . . .
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . .
109
Hysteresis Modeling
110
. . . . . . . . . . . . . . . . . . . . . . . . . . .
.. 111
The Classical Scalar Preisach Hysteresis Model . . . . . . .
112
3.3.1
Graphical Interpretation of the CSPM
. . . . . . .
114
3.3.2
Identification and Implementation of the CSPM . .
118
3.2
P rior A rt
3.3
.
.
.
.
The Chua Hysteresis Model
3.4.1
133
Identification and Implementation of the Chua Model
134
Sum m ary
. . . . . . . . . . . . . . . . .
135
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Reluctance Actuator Hysteresis Modeling
137
4.1
.
3.5
.
Introduction to Hysteresis
.
. . . . . . . . . . . . . . . . . .
3.1
3.4
Basic Reluctance Actuator Model . . . . . . . . . . . . . . . . . .
138
4.1.1
Incorporating the Hysteresis Model into the Actuator Model
139
4.1.2
Hysteresis Model Parameters
140
8
. . . . . . . . . . . . . . . .
.
4
.
.
Current-Biased Linearization . . . . . . . . . . . . .
.
3
2.4.1
.
2.6
71
.
2.5
Linearization via Actuator Configuration . . . . . . . . . .
.
2.4
142
Analysis of Errors from Hysteresis . . . . . . . . . . . . . . . . . . .
143
4.2.1
Offset Error . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
4.2.2
Hysteresis Error . . . . . . . . . . . . . . . . . .
150
.
.
.
4.2
Current Profile
.
. . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3
Analysis of Error from Gap Disturbance
. . . . . . . .
155
4.4
Reluctance Actuator Model with Flux Feedback . . . .
159
.
.
4.3
Stiffness with Flux and Current Control
. . . .
160
4.4.2
Stiffness with Flux Control and Voltage Drive .
165
4.4.3
Comparison between Compensated Stiffness with and without
.
.
4.4.1
. . . . . . .
169
Augmenting the Reluctance Actuator Model . . . . . .
. . . . . . .
172
4.6
. . . . . . . . . . . . . . . .
. . . . . . .
172
4.5.2
Fringing Flux . . . . . . . . . . . . . . . . . . .
. . . . . . .
174
4.5.3
Electrical Modeling . . . . . . . . . . . . . . . .
. . . . . . .
176
4.5.4
Position Control . . . . . . . . . . . . . . . . . .
. . . . . . .
176
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
177
.
.
.
Sum m ary
181
Eddy Current and Power Dissipation Modeling
Diffusion Equation
. . . . . . . . . . . . . . . . . . . . . . . . .
182
5.2
Skin Frequency and Skin Depth . . . . . . . . . . . . . . . . . .
182
5.3
Classical Eddy Currents
. . . . . . . . . . . . . . . . . . . . . .
184
Effect of Gap Disturbances on Classical Eddy Currents .
189
. . . . . . . . . . . . . . . . . . . . . . .
190
. .
196
Power Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . .
199
.
.
.
Hysteresis Loss
. . . . . . . . . . . . . . . . . . . . . . .
200
5.5.2
Classical Eddy Current Loss . . . . . . . . . . . . . . . .
202
5.5.3
Excess Eddy Current Loss . . . . . . . . . . . . . . . . .
204
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205
Reluctance Actuator Modeling for Real-Time Flux Estimation
209
5.6
.
.
5.5.1
.
5.5
Excess Eddy Currents in 50%Ni-50%Fe Laminations
.
5.4.1
.
Excess Eddy Currents
.
5.4
.
5.1
5.3.1
6
.
Angle Disturbance
Sum m ary
.
5
4.5.1
.
4.5
.
.
Current Control . . . . . . . . . . . . . . . . . .
9
6.1
Motivation
6.2
Original Model with Stabilizing Filter . . . . . . . . . . . . . . . . . .
210
6.4
.
6.2.1
Hysteresis Model Identification
. . . . . . . . . . . . . . . . .
211
6.2.2
Analysis of Loop Stability . . . . . . . . . . . . . . . . . . . .
215
6.2.3
Stabilizing Filter Design . . . . . . . . . . . . . . . . . . . . . 217
Flux Estimation Using an Observer . . . . . . . . . . . . . . . . . . .
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.3.2
Simulation Results and Analysis . . . . . . . . . . . . . . . . .
6.3.3
Controller Design with Higher Gain . . . . . . . . . . . . . . . 225
Observer with Adaptive Controller Gain
. . . . . . . . . . . . . . . .
226
Simulation Results
. . . . . . . . . . . . . . . . . . . . . . . .
228
6.4.2
Preliminary Experimental Results . . . . . . . . . . . . . . . .
228
6.4.3
Incorporating a Gap Disturbance . . . . . . . . . . . . . . . .
233
6.4.4
Numeric Problems with the Della Torre Model . . . . . . . . .
234
6.4.5
Hui Hysteresis Model Implementation . . . . . . . . . . . . . .
238
Flux Estimation Using Interpolation
6.6
Observer with Sheared Hysteresis Model
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
240
244
6.6.1
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.6.2
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 247
6.6.3
Incorporating Eddy Currents and Feedforward Control
. . . .
248
Flux Estimation Using the Chua Model . . . . . . . . . . . . . . . . .
250
6.7.1
Hysteresis Model Identification
. . . . . . . . . . . . . . . . .
250
6.7.2
Experimental Results . . . . . . . . . . . . . . . . . . . . . . .
252
6.7.3
Gap Incorporation
254
6.7.4
Chua Model Inversion
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
256
Error Analysis . . . . . . . .
257
6.8.1
Linear Analysis of Observer Response to Gap Disturbance
257
6.8.2
Analysis of Adaptive Gain Observer . . . . . . . . . . . .
259
6.8.3
Analysis of Original Observer
. . . . . . . . . . . . . . .
260
6.8.4
Analysis of SHM Observer . . . . . . . . . . . . . . . . .
261
.
.
.
6.8
223
6.4.1
6.5
6.7
220
6.3.1
.
6.3
. . . . 210
10
6.9
7
262
.................................
265
Magnetic Circuit Modeling of Reluctance Actuators
7.1
Three-pole Actuator Magnetic Circuit Network
. . . . . . . . . . . .
266
7.2
Solving the Magnetic Circuit via Linear Network Theory . . . . . . .
268
7.3
Reluctance Calculations
. . . . . . . . . . . . . . . . . . . . . . . . .
276
7.3.1
Reluctance Calculations for the Air Gap and Core . . . . . . .
277
7.3.2
Reluctance Calculations for Fringing Fields and Leakage Fields
278
7.4
Solving the Magnetic Circuit via Nonlinear Network Theory
7.5
Magnetic Circuit Simulations
7.6
8
Summary ........
. . . . .
285
. . . . . . . . . . . . . . . . . . . . . .
291
7.5.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .
291
7.5.2
Linear Simulations
. . . . . . . . . . . . . . . . . . . . . . . .
291
7.5.3
Nonlinear Simulations
. . . . . . . . . . . . . . . . . . . . . .
293
7.5.4
Nonlinear Simulations with Additional Lumped-parameter Elem ents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
7.5.5
Nonlinear Simulations Using Finite Element Analysis . . . . .
297
7.5.6
Finite Elemenent Analysis of 49% NiFe . . . . . . . . . . . . . 299
Sum mary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
One-DoF Reluctance Actuator Testbed Design
301
303
8.1
Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . .
303
8.2
One-DoF Testbed Concepts
. . . . . . . . . . . . . . . . . . . . . . .
304
8.2.1
Differential-mode Actuation Configuration
8.2.2
Differential-mode Actuation Configuration with Gap Distur-
. . . . . . . . . . .
305
bance Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . .
306
Long-stroke Stage Configuration . . . . . . . . . . . . . . . . .
307
8.3
Testbed Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
307
8.4
Reluctance Actuator Prototype Design
. . . . . . . . . . . . . . . . .
311
8.4.1
Actuator Pole Face Sizing . . . . . . . . . . . . . . . . . . . .
313
8.4.2
Coil Amp-turns . . . . . . . . . . . . . . . . . . . . . . . . . . 313
8.4.3
Coil Window Area . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3
11
314
8.4.5
Selecting Cut Cores and I-bar . . . . .
. . . . . . . . . . . . 315
8.4.6
Simulations of Actuator Design
. . . .
. . . . . . . . . . . . 316
8.4.7
Housing Assembly Design
. . . . . . .
. . . . . . . . . . . . 318
8.4.8
Manufacture and Assembly
. . . . . .
. . . . . . . . . . . . 319
8.4.9
Sense Coil . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 324
.
.
.
.
.
. . . . . . . . . . . . 315
. . . . . . . . . . . . 326
Reluctance Actuator Electrical Characteristics
. . . . . . . . . . . . 326
8.5.1
Resistive Voltage
. . . . . . . . . . . .
. . . . . . . . . . . . 327
8.5.2
Inductive Voltage . . . . . . . . . . . .
. . . . . . . . . . . . 328
8.5.3
Speed Voltage . . . . . . . . . . . . . .
. . . . . . . . . . . . 331
8.5.4
Summary of Electrical Parameters . . .
. . . . . . . . . . . . 332
. .
. . . . . . . . . . . . 332
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 334
.
.
.
.
8.4.10 Summary of Design Parameters.....
Voice Coil Actuator Electrical Parameters
8.7
Sum mary
.
8.6
Loop-Widening Phenomenon
. . . . . . . . . . . . 341
9.1.2
Actuator Simulations . . . . . . . . . .
.
. . . . . . . . . . . . 341
9.1.3
Multiple-pole Sense Coil Measurements
. . . . . . . . . . . .
9.1.4
CoFe Lab Prototype Actuator . . . . .
. . . . . . . . . . . . 345
9.1.5
CoFe ASML Prototype Actuator
. . .
. . . . . . . . . . . . 346
Cut-core Experiments . . . . . . . . . . . . . .
. . . . . . . . . . . . 348
9.2.1
49% NiFe Cut-core . . . . . . . . . . .
. . . . . . . . . . . . 348
9.2.2
CoFe Cut-core . . . . . . . . . . . . . .
. . . . . . . . . . . . 350
Material B-H Experiments . . . . . . . . . . .
. . . . . . . . . . . . 351
9.3.1
50% NiFe . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 352
9.3.2
49% NiFe . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 353
Instrumentation . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 356
9.4.1
. . . . . . . . . . . . 356
.
.
.
.
.
.
.
.
Gaussmeter Measurements . . . . . . .
.
9.4
9.1.1
.
9.3
. . . . . . . . . . . . 338
.
9.2
. . . . . . . . . . . . .
Actuator Experiments
.
9.1
337
Air Core Test Setup
. . . . . . . . . .
.
9
Wire Gauge ...................
.
8.5
8.4.4
12
343
9.5
9.4.2
Current Sense Resistor . . . . . . . . . . . . . . . . . . . . . .
357
9.4.3
AM502 Differential Amplifiers . . . . . . . . . . . . . . . . . .
361
9.4.4
LabVIEW Data Acquisition System . . . . . . . . . . . . . . .
362
9.4.5
AD620 Instrumentation Amplifier Front-End . . . . . . . . . .
362
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
364
Sum m ary
367
10 One-DoF Testbed Experimental Results
. . . . . . . . . . . . . . . . . . . . . . . . . . .
368
. . . . . . . . . . . . . . . . . .
368
10.1.2 Position Controller . . . . . . . . . . . . . . . . . . . . . . . .
371
. . . . . . . . . . . . . . . . . . . . . . . . . . .
373
10.2.1 Flux-Current-Gap Calibration . . . . . . . . . . . . . . . . . .
376
. . . . . . . . . . . . . . . . . . .
378
10.1 Controller Structure
10.1.1 Voice Coil Current Controller
10.2 Actuator Calibration
10.2.2 Force-Flux-Gap Calibration
10.2.3 Voice Coil Calibration
. . . . . . . . . . . . . . . . . . . . . . 381
10.3 SHM Observer Implementation
10.3.1 Calibration
. . . . . . . . . . . . . . . . . . . . .
381
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
10.3.2 Observer Controller Design . . . . . . . . . . . . . . . . . . . .
384
. . . . . . . . . . . . . . . . . . . .
384
10.3.3 Experimental Verification
10.4 Hybrid Flux Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 391
10.5 Flux Control Loop Design . . . . . . . . . . . . . . . . . . . . . . . .
394
10.6 Force Control with Flux Feedback . . . . . . . . . . . . . . . . . . . .
397
10.6.1 Force Frequency Response . . . . . . . . . . . . . . . . . . . .
398
10.6.2 Reluctance Actuator Stiffness . . . . . . . . . . . . . . . . . .
400
10.6.3 Force Pulse Experiments . . . . . . . . . . . . . . . . . . . . .
408
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
413
10.7 Summ ary
417
11 Conclusion and Future Work
11.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
11.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
418
. . . . . . . . . . .
418
11.2.1 Further Investigation into Force Accuracy
11.2.2 Analog or FPGA Flux Control Loop . . . . . . . . . . . . . . 419
13
11.2.3 Optimized Actuator Design
. . . . . . . . . . . . . . . . . . .
419
. . . . . . . . . . . . . . . . . . . . . . . . . .
420
11.2.5 Integration into 6-DoF Stage . . . . . . . . . . . . . . . . . . .
420
11.2.6 Other Applications . . . . . . . . . . . . . . . . . . . . . . . .
420
11.2.4 Cross-Coupling
14
List of Figures
1-1
Diagram of lithography scanner. . . . . . . . . . . . . . . . . . . . .
36
1-2
1-DoF model of reticle stage. . . . . . . . . . . . . . . . . . . . . . .
37
1-3
Schematic diagram of ASML's extended target design invented by
Hol. Figure is taken from US Patent 8,687,171 [42].
1-4
. . . . . . . . .
39
Schematic diagram of ASML's 2-DoF actuator design invented by Hol
showing the coil wiring (top) and 3-D view of the actuator (bottom).
Figure is taken from US Patent 8,472,010 [41]. . . . . . . . . . . . .
1-5
40
Schematic diagram of Nikon's reluctance actuator design invented by
Ono showing curved target. Figure is taken from US Patent 6,906,334
[6 7].
1-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Schematic diagram of precision stage actuated with two reluctance
actuators as a single unit. Figure is taken from US Patent 6,130,517
[82].
1-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Schematic diagram of short-stroke stage actuated by two reluctance
actautors. Figure is taken from US Patent 8,553,205 [75]. . . . . . .
43
1-8
Three-pole reluctance actuator.
. . . . . . . . . . . . . . . . . . . .
46
1-9
Reluctance actuator stator prototype. . . . . . . . . . . . . . . . . .
48
1-10
CAD model of reluctance actuator.
. . . . . . . . . . . . . . . . . .
48
1-11
Photograph of 1-DoF testbed. . . . . . . . . . . . . . . . . . . . . .
49
1-12
LEFT: B-I data measured on reluctance actuator demonstrating
loop-widening phenomenon; RIGHT: zoomed in near origin. ....
15
50
1-13
LEFT: SHM observer real-time flux estimate compared to sense coil
measurement with 35 pm p-p gap disturbance about nominal gap of
530 pm; TOP RIGHT: zoomed in at saturation; BOTTOM RIGHT:
zoom ed in at origin. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-14
50
Measured frequency responses of flux-controlled reluctance actuator
stiffness.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2-1
Flux path for a three-pole reluctance actuator
. . . . . . . . . . . .
55
2-2
Gauss's Law applied to the center pole face air gap. . . . . . . . . .
56
2-3
Maxwell surface for calculating the magnetic force on the mover. . .
59
2-4
The Lorentz force on a current-carrying conductor immersed in an
external magnetic flux density B.
. . . . . . . . . . . . . . . . . . .
61
2-5
Cross-section of a Lorentz actuator. . . . . . . . . . . . . . . . . . .
61
2-6
Ampere's law applied to the Lorentz actuator.
62
2-7
Force density comparison between a Lorentz actuator (left) and a
. . . . . . . . . . . .
reluctance actuator (right) . . . . . . . . . . . . . . . . . . . . . . .
2-8
64
The coercivity (Hc), saturation flux density (B,), and remanence (B,)
are indicated on the major hysteresis loop.
. . . . . . . . . . . . . .
68
2-9
A current-biased reluctance actuator. . . . . . . . . . . . . . . . . .
72
2-10
The flux-biased actuator with a permanent magnet generating the
bias flux. Design presented by Lu [56].
. . . . . . . . . . . . . . . .
74
2-11
A flux-biased actuator with an electromagnet generating the bias flux. 75
2-12
An actuator air gap causes the hysteresis loop to shear. . . . . . . .
2-13
LEFT: reluctance actuator with oversized target.
77
RIGHT: target
w ith teeth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
2-14
LEFT: a single E-lamination. RIGHT: an E-lamination stack.
80
2-15
TOP LEFT: a tape-wound core with cuts in the center. TOP RIGHT:
a tape-wound cut-core showing the lamination orientation.
. . .
BOT-
TOM: two cut-cores placed side-by-side to form a three-pole actuator
stator core.. ...
....
. . ..........
16
..
. . . . . . . ..
. .
81
.
2-16
Block diagrams for four reluctance actuator control strategies.
2-17
Block diagrams for three additional reluctance actuator control strategies.
.
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
. . .
91
2-18
A reluctance actuator with a Hall-effect sensor in the air gap.
2-19
A reluctance actuator with a sense coil wound around the center pole
face.
2-20
92
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A reluctance actuator with strain guages mounted to flexures attached to the target.
2-21
84
. . . . . . . . . . . . . . . . . . . . . . . . . .
95
A reluctance actuator with three piezoelectric load cells connected to
the stator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
2-22
A reluctance actuator with an accelerometer attached to the target.
97
2-23
A reluctance actuator with three piezoelectric load cells and one accelerometer mounted to the stator.
. . . . . . . . . . . . . . . . . .
98
2-24
Block diagram of a controller with feedback linearization. . . . . . .
99
2-25
Feedback linearization for a reluctance actuator.
2-26
Standard ILC configuration.
2-27
LEFT: chuck configuration with one-sided (pull-only) actuators. RIGHT:
. . . . . . . . . . .
100
. . . . . . . . . . . . . . . . . . . . . .
101
chuck configuration with double-sided (push-pull) actuators.
2-28
. . . .
103
LEFT: Target is preloaded with spring against hard stops for normal
operation. RIGHT: Spring allows target to function as soft stop in
case of crash.
2-29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
A three-pole actuator showing the effect on flux distribution of a
relative rotation between the stator and target . . . . . . . . . . . .
106
2-30
A three-pole actuator with two independent actuator coils.
. . . . .
106
2-31
A five-pole actuator with two independent actuator coils. . . . . . .
108
3-1
Hysteresis phenomenon, shown as magnetization versus applied field. 111
3-2
A Preisach hysteron with the threshold values a and 3 for switching
output states. ......
3-3
..................
A parallel connection of weighted hysterons.
17
..........
.113
. . . . . . . . . . . . .
114
3-4
The Preisach plane. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-5
A: Input u(t) increased -u,
115
to u 1 . B: Input u(t) decreased from u1
to u 2 . C: Input u(t) increased from U2 to u 3 . D: Input u(t) increased
from U 3 to u 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-6
117
LEFT: A possible instantiation of the Preisach plane subdividing line.
RIGHT: An impossible instantiation of the Preisach plane subdivid-
ing line.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
3-7
Virgin curve and major loop of the hysteresis curve. . . . . . . . . .
120
3-8
The Preisach plane showing the demagnetized state. . . . . . . . . .
122
3-9
LEFT: Preisach plane with H increased from the demagnetized state.
RIGHT: Preisach plane with H decreased from the demagnetized state. 123
3-10
Three hysteresis loops constructed from the Della Torre CSPM with
different values of Hc. . . . . . . . . . . . . . . . . . . . . . . . . . .
126
3-11
The dM/dH values needed to evaluate a-. . . . . . . . . . . . . . . .
127
3-12
Three hysteresis loops constructed from the Della Torre CSPM with
different values of a-.
3-13
. . . . . . . . . . . . . . . . . . . . . . . . . .
127
Minor loops constructed from the Della Torre implementation of the
CSPM with parameters H, = 5 A/m, a = 20, poM, = 1.5 T. ....
128
3-14
The area of the Preisach plane corresponding to T(H1 , H2 ).
129
3-15
LEFT: Preisach plane corresponding to the descending branch of ma-
....
jor hysteresis loop. RIGHT: Preisach plane corresponding the ascending branch of the major hysteresis loop. . . . . . . . . . . . . . . . .
3-16
129
Major loop and minor loops constructed from the Hui implementation
of the CSPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
f
132
3-17
Identification of
for the Chua model.
. . . . . . . . . . . . . . . .
134
3-18
Identification of g for the Chua model.
. . . . . . . . . . . . . . . .
135
3-19
Major loop and minor loops constructed from the frequency-independent
4-1
Chua m odel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
136
Block diagram for actuator model. . . . . . . . . . . . . . . . . . . .
138
18
4-2
Block diagram for actuator model with Della Torre hysteresis model.
140
4-3
Major loop of hysteresis model used for simulation.
. . . . . . . . .
141
4-4
TOP: Acceleration profile for simulation. BOTTOM: Associated current profile for simulation of single reluctance actuator.
. . . . . . .
142
. . . . . . . . . . . . . . . . . . . . . .
144
4-5
Hysteresis offset error ABff.
4-6
ABmax is approximately equal to AB..
. . . . . . . . . . . . . . . .
145
4-7
Actuator with flux path shown and dimensions marked. . . . . . . .
147
4-8
Simulation of a reluctance actuator showing the offset force due to
hysteresis.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
150
4-9
AFh between the ascending and descending branches. . . . . . . . .
151
4-10
A graphical representation of the relation between the B-I curve and
Ha and Hd.
4-11
153
Simulated force versus current showing the maximum force error from
hysteresis.
4-12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
LEFT: Comparison of AFh from (4.27) with AF from simulation.
RIGHT: Comparison of AFh from (4.28) with AF from simulation.
155
4-13
Simulated gap disturbance. . . . . . . . . . . . . . . . . . . . . . . .
158
4-14
TOP: Simulated actuator force output with and without gap disturbance; BOTTOM: Simulated force error generated by gap disturbance. 159
4-15
Block diagram of a reluctance actuator model with flux feedback.
4-16
Block diagram of reluctance actuator flux control loop with gap disturbance........
4-17
.
..................................
161
Simulated frequency responses of reluctance actuator gap disturbance
rejection with flux and current control for varying flux levels. . . . .
4-18
162
Simulated frequency responses of reluctance actuator gap disturbance
rejection with flux and current control for different nominal air gaps.
4-19
160
163
Simulated frequency response of reluctance actuator gap disturbance
rejection compared with theoretical approximations. . . . . . . . . .
164
4-20
Reluctance actuator with angle disturbance.
. . . . . . . . . . . . .
173
4-21
Block diagram of a reluctance actuator with angle disturbance. . . .
174
19
4-22
Block diagram of a reluctance actuator with fringing fields modeled.
4-23
Block diagram of a reluctance actuator model with the electrical dom ain m odeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-24
I (right).
. . . . . . . . . . . . . . . . .
184
Block diagram for actuator model including the classical eddy current
term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
177
Magnetic field distribution within a single lamination (left) and approximation with uniform
5-2
176
A 2-DoF model of the short-stroke stage with four reluctance actuator
models and gap and angle disturbances. . . . . . . . . . . . . . . . .
5-1
175
187
Comparison of AFeI from (5.21) with AFd1 from simulation. Red line
is a scaled force profile to provide a time reference . . . . . . . . . .
189
5-4
Simulated Ha, Hh, and Hc1 .
190
5-5
Block diagram for actuator model including excess eddy current term. 192
5-6
Comparison of AFddy = AFc,
AFddy
from simulation.
. . . . . . . . . . . . . . . . . . . . . .
+ AFexc from (5.21) and (5.35) with
. . . . . . . . . . . . . . . . . . . . . . . .
195
5-7
Simulated Ha, Hh, He1 , and Hexc using (5.27) for Hexc.
. . . . . . .
196
5-8
Loop widening from eddy currents for the simulated B-H curve. . .
197
5-9
Loop widening from eddy currents for simulated B-I curve. . . . . .
197
5-10
Simulation of
AFeddy
= AFc,+ AFexc using (5.38) for the excess eddy
currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
198
5-11
Simulated Ha, Hh, He1 , and Hrc using (5.38) for Hexc.
199
5-12
A generic acceleration profile for a scanning lithography stage.
5-13
A rectangle of width 2H, and height Bmax approximates the energy
per volume lost from hysteresis.
6-1
. . . . . . .
. . .
. . . . . . . . . . . . . . . . . . . .
211
B-H major loop derived from experimental data and Della Torre
model fit.
6-3
201
Experimental B-I curve and its linear approximation for an actuator
gap of 500 pm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-2
201
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Air gap data for experimental B-H data from Figure 6-2. . . . . . .
20
214
6-4
B-H experimental major loop with no compensation for changing gap. 214
6-5
B-H major loop derived from experimental data and Della Torre
m odel fit.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215
6-6
A simple relay as an approximation for the hysteresis operator.
.
.
216
6-7
Reluctance actuator model with stabilizing filter for real-time.
.
.
218
6-8
LEFT: Simulated B-I curve from Figure 6-7 model compared with
experimental B-I curve; RIGHT: Corresponding simulated B-H curve. 220
6-9
Bode plot of stabilizing filter F(s).
. . . . . . . . . . . . . . . . . .
221
6-10
Block diagram of observer for estimating actuator flux density. . . .
221
6-11
Simulated B-I curve from observer with controller C(s) = -3. Note
errors where dB/dH is small.
6-12
. . . . . . . . . . . . . . . . . . . . .
224
Simulated input current (I) and estimated output current (I) from
observer with controller C(s) = 3-'. . . . . . . . . . . . . . . . . . .
224
. . .
226
6-13
Simulated B-I curve from observer with controller C(s) = 9.
S
6-14
TOP: Simulated input current (I) and estimated output current (I)
from observer with controller C(s) = !-; BOTTOM: Error between
I and i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
. . . . . . . . . . . . . .
6-15
Simulink model of adaptive gain observer.
6-16
Simulated adaptive gain observer estimate of flux density versus input
current.
6-17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
229
230
TOP: Simulated input current (I) and estimated output current (i)
from observer with adaptive gain controller; BOTTOM: Error between I and i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
230
6-18
Clamped actuator with sense resistor shown. . . . . . . . . . . . . .
231
6-19
LEFT: Real-time observer estimates of flux density versus measured
current; RIGHT: Real-time observer B-H curves.
. . . . . . . . . .
232
6-20
Real-time observer $-H loop showing 'whiskers' near reversal points. 232
6-21
Measured actuator current filtered at 200 Hz showing high-frequency
reversals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
233
6-22
LEFT: Real-time observer estimate of flux density versus measured
current with simulated gap disturbance; RIGHT: Real-time observer
B-H curves with simulated gap disturbance.
. . . . . . . . . . . . .
234
6-23
The effect of a gap disturbance on the observer estimated flux density.234
6-24
Comparison between observer B-H loops for different gap disturbances. 235
6-25
Comparison of different numeric solvers for Della Torre hysteresis
model for same H input.
. . . . . . . . . . . . . . . . . . . . . . . .
6-26
Simulated input current and gap step to actuator model.
6-27
Simulated B-H curve showing the effect of a -200 pm on the Della
. . . . . .
Torre hysteresis model. . . . . . . . . . . . . . . . . . . . . . . . . .
6-28
236
236
237
LEFT: Simulated Hui-model observer estimate of flux density versus
input current with twice original gap disturbance; RIGHT: Simulated
Hui-model observer B-H curve with twice original gap disturbance.
6-29
239
LEFT: Simulated Hui-model observer estimate of flux density versus
input current with five times the original gap disturbance; RIGHT:
Simulated Hui-model observer B-H curve with five times the original
gap disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
240
6-30
Interpolation scheme for estimating actuator flux density. . . . . . .
241
6-31
Simulation with gap disturbance:
comparison of B-I curves using
gap interpolation scheme (blue) and adaptive-gain observer with gap
disturbance applied directly (green). . . . . . . . . . . . . . . . . . .
6-32
AB between interpolation model simulation and direct gap model
simulation both with original gap disturbance.
6-33
. . . . . . . . . . . .
242
Comparison between interpolation method and direct gap method.
(Simulation).
LEFT: twice the original gap disturbance; RIGHT:
five times the original gap disturbance.
6-34
242
. . . . . . . . . . . . . . . .
243
The relationship between g2 , g and goff is shown on the reluctance
actuator........
..................................
244
6-35
The variable H2 is obtained by adding
6-36
Observer block diagram with SHM . . . . . . . . . . . . . . . . . . . 247
22
292f
62Fe
to H. . . . . . . . . . .
245
6-37
Simulation with gap disturbance: comparison of B-I curves using
sheared-model observer (blue) and adaptive-gain observer (green). .
6-38
248
Estimated flux density versus input current simulated for SHM observer with different gap disturbances (Ix, 2x, 5x, and 10x original
gap disturbance).
6-39
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulink model of sheared-model observer with eddy current effects
. . . . . . . . . . . . . . . . . . . .
251
functions for Chua model. . . . . . . . . . . . . . . . .
252
and feedforward path included.
6-40
f -1
6-41
Frequency-independent
and G-
Chua model B-I major loop output com-
pared to measured B-I data. . . . . . . . . . . . . . . . . . . . . . .
6-42
254
Frequency-dependent Chua model B-I major loop output compared
to measured B-I data.
6-44
253
Frequency-independent Chua model B-I minor loop output compared to measured B-I data. . . . . . . . . . . . . . . . . . . . . . .
6-43
249
. . . . . . . . . . . . . . . . . . . . . . . . .
255
Frequency-dependent Chua model B-I minor loop output compared
to measured B-I data.
. . . . . . . . . . . . . . . . . . . . . . . . .
256
. . . . . . . . . . . . . . . . .
258
6-45
Error integration over one time step.
6-46
AH going from low-dB/dH region to high-dB/dH region.
7-1
Magnetic circuit for the half-actuator. . . . . . . . . . . . . . . . . .
266
7-2
Magnetic circuit for the half-actuator in standard form. . . . . . . .
267
7-3
The magnetic circuit with the magnetic potentials (4ys) at each node
and the magnetic fluxes (<bi 8 ) through each reluctance.
7-4
. . . . . 260
. . . . . . .
The half-actuator with dimensions labeled. The actuator core has a
depth d, into the paper. The target has a depth da into the paper. .
7-5
268
278
Numbered flux tubes are shown representing probable flux paths for
the electromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
7-6
Figure from [74] showing the 3-dimensional shapes of the flux tubes.
280
7-7
Flux paths 2 and 4 showing characteristic dimensions. . . . . . . . .
281
7-8
Flux paths 3 and 5 showing characteristic dimensions. . . . . . . . .
282
23
. . . . . . 284
7-9
The leakage flux tube showing characteristic dimensions.
7-10
Magnetic circuit for example Tableau analysis. . . . . . . . . . . . . 286
7-11
Test fixture for measuring electromagnetic actuator characteristics. .
292
7-12
Sample electromagnet for testing.
. . . . . . . . . . . . . . . . . . .
292
7-13
Simulated and measured force at 50 pim gap. Linear magnetic circuit
used for simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
7-14
Simulated and measured force at 450 pm gap. Linear magnetic circuit
used for sim ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
294
7-15
Experimentally measured B-H data for a 50-50% NiFe toroidal core.
294
7-16
Single-valued B-H function (red) computed from measured B-H (blue).295
7-17
Simulated and measured force at 50 lm gap. Simulated force includes
nonlinear core reluctance elements.
. . . . . . . . . . . . . . . . . .
296
7-18
Augmented magnetic circuit. . . . . . . . . . . . . . . . . . . . . . .
296
7-19
Simulated and measured force at 50 jim gap. Augmented nonlinear
. .
simulation (green) uses additional lumped parameters in model.
297
7-20
COMSOL FEA model of actuator . . . . . . . . . . . . . . . . . . . 298
7-21
FEA simulation and measured force for 50% NiFe at 50 lpm gap.
. .
298
7-22
49% NiFe sample used to measure B-H curve. . . . . . . . . . . . .
299
7-23
Experimentally measured B-H data for a 49% NiFe toroidal core
300
7-24
B-H data from Vacuumschmelze for 49% NiFe and 50% NiFe.
7-25
FEA and measured force at 50 jim gap. Simulated force used 49%
.
.
.
300
NiFe B-H data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
8-1
Testbed concept with differential-mode actuation.
8-2
Testbed concept with differential-mode actuation and gap disturbance stage. .......
. . . . . . . . . .
305
...............................
306
8-3
Testbed concept with long-stroke stage. . . . . . . . . . . . . . . . .
308
8-4
CAD model of 1-DoF testbed. . . . . . . . . . . . . . . . . . . . . . 308
8-5
Photograph of 1-DoF testbed. . . . . . . . . . . . . . . . . . . . . .
309
8-6
CAD model of cross-sectional view of 1-DoF testbed.
309
24
. . . . . . . .
8-7
Testbed system diagram. . . . . . . . . . . . . . . . . . . . . . . . .
311
8-8
Reluctance actuator stator prototype. . . . . . . . . . . . . . . . . .
312
8-9
Coil window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
8-10
Dimensions of c-core. Figure taken from Magnetic Metals website. .
315
8-11
Dimensions of I-bar . . . . . . . . . . . . . . . . . . . . . . . . . . .
316
8-12
Magnetic circuit simulation of force versus current density for reluctance actuator design . . . . . . . . . . . . . . . . . . . . . . . . . .
316
. . . . .
317
8-13
FEA model of reluctance actuator prototype at saturation.
8-14
Concept of target design with top corners chamfered to reduce mass. 317
8-15
FEA simulation of force versus current density for reluctance actuator
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
318
8-16
CAD model of reluctance actuator housing. . . . . . . . . . . . . . .
319
8-17
Stator housing.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
320
8-18
Actuator stator before potting . . . . . . . . . . . . . . . . . . . . .
320
8-19
Potted actuators.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
321
8-20
Hot melt glue applied to cut cores.
. . . . . . . . . . . . . . . . . .
321
8-21
Photograph of pole face laminations frayed outward from grinding. .
322
8-22
Fixture for making c-cores parallel to housing surface. . . . . . . . .
322
8-23
Photograph of fixture for making c-cores parallel to housing surface.
323
8-24
LEFT: Target housing; CENTER: Housing with I-bar in pocket;
prototype.
RIGHT: Mold for potting target assembly.
. . . . . . . . . . . . . .
323
. . . . . . . . . . . . .
324
8-25
Target assembly showing potting procedure.
8-26
Completed target assembly.
. . . . . . . . . . . . . . . . . . . . . .
325
8-27
Hand-wound sense coil. . . . . . . . . . . . . . . . . . . . . . . . . .
325
8-28
Circuit diagram of actuator setup. . . . . . . . . . . . . . . . . . . .
326
8-29
Circuit model of reluctance actuator.
. . . . . . . . . . . . . . . . .
329
8-30
Frequency response of input amplifier voltage to reluctance actuator
current for different bias currents. . . . . . . . . . . . . . . . . . . .
8-31
330
Crown amplifier VI limits. Figure taken from [30]. . . . . . . . . . . 335
25
9-1
LEFT: B-I data measured on reluctance actuator demonstrating
loop-widening phenomenon; RIGHT: zoomed in near origin. ....
.338
9-2
Measured B-F curve showing that the force leads the flux density. .
9-3
LEFT: F-I data measured on reluctance actuator; RIGHT: zoomed
in near 2 A .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-4
Clamped actuator setup.
9-5
Instrumentation diagram for actuator experiments.
9-6
LEFT: Measured B-I curves with B estimated from sense coil; RIGHT:
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
339
339
340
341
Measured B-I curves with B measured by Gaussmeter. Bottom plots
are zoomed-in graphs of the top plots. . . . . . . . . . . . . . . . . .
9-7
LEFT: Simulated B-I curves at different frequencies; RIGHT: zoomed
in near origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-8
342
343
LEFT: Measured B-I curves estimated from center pole face sense
coil; RIGHT: Measured B-I curves estimated from outer pole face
sense coil. Bottom plots are zoomed-in graphs of the top plots. . . .
344
9-9
CoFe actuator prototype stator.
346
9-10
LEFT: Measured A-I curves from CoFe lab prototype actuator; RIGHT:
Zoomed in near origin.
9-11
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
346
LEFT: Measured B-I curves from ASML CoFe prototype actuator
at 500 pm gap; RIGHT: Measured B-I curves from ASML CoFe prototype actuator at 1 mm gap. Bottom plots are zoomed-in graphs of
the top plots.
9-12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
349
Photograph of CoFe cut-core with machine-wound primary coil on
left and hand-wound secondary sense coil on right. . . . . . . . . . .
9-15
349
LEFT: Measured A-I curves from 49% NiFe cut-core; RIGHT: Zoomed
in near origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-14
347
Experimental setup for measuring B-I characteristics of single 49%
NiFe cut-core.
9-13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
351
LEFT: Measured A-I curves from CoFe cut-core; RIGHT: Zoomed in
near origin.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
351
9-16
Photograph of 50% NiFe toroid with bifilar windings.
. . . . . . . .
352
9-17
Measured B-H curves on 50% NiFe toroid specimen.
. . . . . . . .
353
9-18
LEFT: Measured B-H loop on 50% NiFe toroid driven at 1 Hz;
RIGHT: Measured sense coil voltage v.. . . . . . . . . . . . . . . . .
353
9-19
49% NiFe sample used to measure B-H curves. . . . . . . . . . . . .
354
9-20
Measured B-H curves on 49% NiFe specimen.
. . . . . . . . . . . .
355
9-21
LEFT: 49% NiFe cut-core used for testing B-H properties; RIGHT:
Experimental setup for testing B-H properties . . . . . . . . . . . .
355
. . . . . . . . . . . .
356
9-22
Measured B-H curves on 49% NiFe cut-cores.
9-23
Photograph of 'air core' with primary and secondary coils.
. . . . .
357
9-24
Experimental setup for testing air core. . . . . . . . . . . . . . . . .
358
9-25
LEFT: Measured A-I curves on air core with 0.01 Q sense resistor;
RIGHT: Measured A-I curves on air core with 0.1 Q sense resistor.
Bottom plots are zoomed-in graphs of the top plots. . . . . . . . . .
9-26
359
LEFT: Measured A-I loop on air core; RIGHT: Measured A-I loop
. .
360
9-27
Experimental setup for testing air core. . . . . . . . . . . . . . . . .
363
10-1
Controller configuration used for 1-DoF testbed. . . . . . . . . . . .
369
10-2
Measured frequency response for uncompensated voice coil actuator.
370
10-3
Predicted and measured frequency responses for voice coil loop trans-
on air core with AM502 differential amplifier channels switched.
m ission.
10-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
370
Measured frequency response from voice coil current to air bearing
position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371
10-5
Air bearing setup with preload springs. . . . . . . . . . . . . . . . .
372
10-6
Air bearing frequency responses: P - plant (measured); C - controller
(predicted); LT - loop transmission (measured).
10-7
10-8
. . . . . . . . . . .
373
Controller configuration used for initial calibration of reluctance actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
374
Air gap fluctuation during calibration at go = 500 pm. . . . . . . . .
374
27
10-9
Controller configuration used for calibration of reluctance actuator
with voice coil feedforward added. . . . . . . . . . . . . . . . . . . .
375
10-10 Air gap fluctuation during calibration with force feedforward at go =
500 pm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
10-11 B-I curves at different operating gaps for reluctance actuator calibration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
377
10-12 B(Ir, g) lookup table. . . . . . . . . . . . . . . . . . . . . . . . . . . 377
10-13 Mutual inductance Lm of reluctance actuator as a function of gap g.
378
10-14 F-B curves at different operating gaps for reluctance actuator calibration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
379
10-15 LEFT: F(B, g) lookup table; RIGHT: Inverse B(F, g) lookup table.
379
10-16 Reluctance actuator k-factor . . . . . . . . . . . . . . . . . . . . . .
380
10-17 Plot of voice coil current versus reluctance actuator load cell measurements used to estimate Kmc.
. . . . . . . . . . . . . . . . . . .
382
10-18 Observer block diagram with SHM. . . . . . . . . . . . . . . . . . .
382
10-19 B-I data at 530 pm gap used to calibrate SHM.
383
. . . . . . . . . . .
10-20 LEFT: major loop of SHM observer real-time flux estimate compared
to sense coil measurement at nominal gap of 530 pm; TOP RIGHT:
zoomed in at saturation; BOTTOM RIGHT: zoomed in at origin.
.
385
10-21 Left: minor loop of SHM observer real-time flux estimate compared to
sense coil measurement at nominal gap of 530 pm; Top right: zoomed
in at maximum flux density; Bottom right: zoomed in at origin.
. .
386
10-22 LEFT: SHM observer real-time flux estimate compared to sense coil
measurement at gap of 580 pm; TOP RIGHT: zoomed in at satura-
tion; BOTTOM RIGHT: zoomed in at origin.
. . . . . . . . . . . . 387
10-23 Encoder measurement showing reluctance actuator stator expansion
due to actuator heating.
. . . . . . . . . . . . . . . . . . . . . . . .
388
10-24 Reluctance actuator driven with constant voltage and air bearing
slide clamped to stator. LEFT: reluctance actuator current measurement. RIGHT: encoder measurement. . . . . . . . . . . . . . . . . .
28
389
10-25 LEFT: SHM observer real-time flux estimate compared to sense coil
measurement with sinusoidal gap disturbance about nominal gap
of 530 pm; TOP RIGHT: zoomed in at origin; BOTTOM RIGHT:
. . . . . . . . . . . . . . . . . . . . . . . .
390
10-26 Gap disturbance applied to air bearing stage. . . . . . . . . . . . . .
390
zoomed in at saturation.
10-27 LEFT: SHM observer real-time flux estimate compared to sense coil
measurement with scanning gap disturbance about nominal gap of
530 1pm; TOP RIGHT: zoomed in at saturation; BOTTOM RIGHT:
zoom ed in at origin. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
392
. . . . . . . . . . . . . . . . . .
394
. . . . . . . .
396
10-28 Complementary filter structure for estimating actuator flux.
10-29 Block diagram of flux control loop.
391
10-30 Measured frequency response for reluctance actuator.
-
10-31 Reluctance actuator frequency responses: P - plant (measured); C
controller (predicted); LT - loop transmission (measured) . . . . . .
397
10-32 Block diagram for reluctance actuator force control. . . . . . . . . .
398
10-33 Measured flux-controlled reluctance actuator frequency responses from
desired force to measured force.
. . . . . . . . . . . . . . . . . . . .
399
10-34 Measured flux-controlled reluctance actuator frequency responses from
desired force to measured force for frequencies below 100 Hz.
. . . .
400
10-35 Measured frequency responses of flux-controlled reluctance actuator
stiffness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
401
10-36 Measured frequency responses of flux-controlled reluctance actuator
gap disturbance rejection at different flux levels. . . . . . . . . . . .
403
10-37 Measured frequency responses of flux-controlled reluctance actuator
gap disturbance rejection for different nominal air gaps. . . . . . . . 404
10-38 Measured frequency responses of flux-controlled reluctance actuator
stiffness at different nominal air gaps. . . . . . . . . . . . . . . . . .
405
10-39 Force pulse experimental results with flux feedback and force-flux
inversion. TOP: reference force profile and measured force profile.
BOTTOM: error between reference force and measured force. . . . . 410
29
10-40 Air gap measurement during force pulse experiment. . . . . . . . . .
411
10-41 Force pulse experimental results with complementary filter pair cutoff
frequency set to 10 Hz. TOP: reference force profile and measured
force profile. BOTTOM: error between reference force and measured
force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-42 Gap disturbance during force pulse experiment.
. . . . . . . . . . .
4 12
413
10-43 Force pulse experimental results with gap disturbance applied. TOP:
reference force profile and measured force profile. BOTTOM: error
between reference force and measured force.
30
. . . . . . . . . . . . .
414
List of Tables
2.1
Comparison of key parameters for several core materials
. . . . . . .
68
4.1
Della Torre model parameters . . . . . . . . . . . . . . . . . . . . . .
140
4.2
Error sources for a reluctance actuator
. . . . . . . . . . . . . . . . .
178
4.3
Reluctance actuator stiffnesses . . . . . . . . . . . . . . . . . . . . . .
179
5.1
Errors from eddy currents in a reluctance actuator . . . . . . . . . . .
206
5.2
Reluctance actuator ferromagnetic power dissipation sources
. . . . .
207
8.1
Reluctance actuator prototype design parameters.
. . . . . . . . . . .
327
8.2
Reluctance actuator prototype electrical parameters.
8.3
Predicted peak electrical variables for reluctance actuator prototype.
333
8.4
Voice coil actuator electrical parameters.
. . . . . . . . . . . . . . . .
333
8.5
Predicted peak electrical variables for voice coil actuator. . . . . . . .
334
9.1
Phase between simulated actuator current and flux density . . . . . .
343
9.2
Phase between actuator current and sense coil voltage . . . . . . . . .
345
9.3
Phase between CoFe lab prototype actuator current and sense coil voltage347
9.4
Phase between actuator current and sense coil voltage for ASML CoFe
prototype actuator. .....
.....
. . ..........
. . . . . . . . .
....
..
. ..
9.5
Phase between 49% NiFe cut-core current and sense coil voltage . . .
9.6
Phase between air core primary coil current and secondary sense coil
voltage with different current sense resistors
9.7
. . . . . . . . . . . . . .
332
348
350
360
Comparison of phase between actuator current and sense coil voltage
for different sense coil differential amplifier gain settings.
31
. . . . . . .
361
9.8
Phase between reluctance actuator current and sense coil voltage using
LabVIEW DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9
362
Phase between reluctance actuator current and sense coil voltage using
AD620 front-end. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
364
Chapter 1
Introduction
In the semiconductor industry, photolithography is a critical part of the process used
to produce integrated circuits (IC's). In a state-of-the-art photolithography scanner, the reticle stage must accelerate at tens of g's while maintaining sub-nanometer
position accuracy. As the industry continues to advance to higher throughputs and
smaller feature sizes, the required accelerations must increase and the required position accuracy must improve. The present state-of-the-art Lorentz-force actuators
used to drive the reticle stage will be unable to achieve these higher accelerations.
Thus, a new actuation technology must be developed. In this thesis, we explore the alternative technology of reluctance actuators. Reluctance actuators can achieve much
higher force densities (force-to-moving-mass ratio) than Lorentz actuators at small air
gaps. Reluctance actuators, however, are significantly more nonlinear than Lorentz
actuators, and so present a challenge in achieving the required position accuracy. This
thesis examines how to model and control a reluctance actuator in order sufficiently
to linearize it for photolithography applications. In this chapter, we first provide a
background of the problem and then present prior art for linearizing reluctance actuators for lithography. We then give an overview of the thesis and summarize the
primary contributions.
33
1.1
Background
In a lithography machine, the desired IC pattern is optically transferred onto a silicon
wafer via a 'reticle' mask that contains a slit for exposing light on the wafer.
A
simplified diagram of the photolithography scanner is shown in Figure 1-1. Both the
silicon wafer and the reticle are scanned back and forth relative to each other so that
the entire silicon wafer can be patterned. The wafer stage is alternately scanned in
y and stepped in x. The reticle stage is scanned in y and is coordinated with the
wafer movement. A scan profile consists of acceleration and deacceleration phases at
the beginning and end of the scan. In the middle of the scan is a constant velocity
phase during which the wafer exposure occurs. More details on the operation of a
lithography machine can be found in [18].
Illumination source
reticle
scanning (y)
reticle stage
Lens
wafer
step (x), scan (y)
wafer stage
Figure 1-1: Diagram of lithography scanner.
Figure 1-2 shows a one-degree-of-freedom (DoF) model of a reticle stage (the wafer
stage is similar). It consists of a long-stroke stage for coarse positioning and a shortstroke stage for fine positioning. The short-stroke stage is magnetically levitated in all
six DoF and is servoed to the desired position referenced from an isolated metrology
frame (yss). The long-stroke stage follows the short-stroke stage by attempting to
34
maintain a constant Ydiff. Lorentz actuators are currently used for the short-stroke
actuation (Fss). Because the features printed on the IC can be as small as tens of
nanometers, the position accuracy of the reticle and wafer during exposure must be
accurate to a few nanometers or better. The servo bandwidth is limited by stage
dynamics, so the majority of the position accuracy comes from the force feedforward
accuracy, which must be better than 99.9% accurate for a typical trajectory. Complicating issues is the long-stroke tracking error, which introduces force disturbances
into the short-stroke stage owing to the non-zero stiffness of the actuator. The maximum allowable actuator stiffness is 0.O001Fm
N/pm for a typical long-stroke peak
tracking error of 10 pm, where Fma is the maximum force output of the actuator.
Yss
Metro
Frame
Reticle
Ydiff
Figure 1-2: 1-DoF model of reticle stage.
To obtain higher throughputs, the reticle and wafer stages must be operated at
higher speeds and accelerations.
or lower moving masses.
These higher accelerations require higher forces
Presently, Lorentz actuators are near their limit in the
acceleration they can achieve. For the scale of air gaps envisioned between actuator
to
stator and payload (-1 mm), reluctance actuators can generate equivalent forces
a Lorentz actuator but with a much smaller actuator moving mass, or alternatively,
they can generate much higher forces with an equivalent actuator moving mass.
Reluctance actuators come with a host of challenges not present in a Lorentz actuator, however. Chief among these is the reluctance actuator nonlinearity. Lorentz
actuators generate a force that is approximately linear with current and have approximately zero stiffness in the driving direction. These features are conducive to
achieving the high accuracy required for the reticle and wafer stages. Reluctance
35
actuators, in contrast, generate a force that is quadratic with current and display a
large negative stiffness. Moreover, they exhibit hysteresis and can also be affected by
eddy currents. These features must be properly compensated for in order for reluctance actuators to achieve the required force accuracy. The main focus of this thesis
is developing ways to model and control the reluctance actuator to force-linearize it
for use with a photolithography scanner. The work has been a collaboration with
ASML, the world's leader in semiconductor lithography machines.
1.2
Prior Art
Reluctance actuators are used in a variety of precision engineering applications, most
notably as magnetic bearings [37]. To the best of our knowledge, however, the lithography application is the only one in which reluctance actuators have been investigated
for achieving high dynamic force accuracy (99.9% or better) and very low stiffness. In
the following subsections, we review the solutions for linearizing reluctance actuators
for use in lithography scanners in the available scholarly literature and patent literature. The prior art references can be categorized into two groups according to their
approach to linearizing the actuator: actuator configuration (e.g., physical geometry,
alternate ways to drive the actuator) and control. These two approaches are often
complementary, i.e., they can be used together.
1.2.1
Actuator Configuration
In [42], Hol of ASML designed a reluctance actuator with a target surface that extends
beyond the surface spanned by the stator pole faces. The target is also called the
mover or armature. A sketch of the actuator is shown in Figure 1-3. The driving
direction is along the y-axis. A current is driven through the coils (340.1 and 340.2)
to generate a magnetic flux in the air gaps (390). This flux generates a force (F) on
the target (310). The target surface (312) has a greater length (LI) in the x-direction
than the length (L2) spanned by the stator (320) pole faces.
Likewise, in the z-
direction, the target surface has a greater length (L3) than the pole face dimension
36
(L4).
Such a configuration makes the reluctance actuator less sensitive to relative
motions between the stator and target in the x- and z-directions.
300
3
-----
340.1
320
-------------
320
.2
,
3 2
L4
39
31
2
12
F,
------------------
1
y
L3
310
310
z
x
Figure 1-3: Schematic diagram of ASML's extended target design invented by Hol.
Figure is taken from US Patent 8,687,171 [42].
Hol also designed a 2-DoF reluctance actuator in [41] that can generate a controllable torque in addition to a controllable force. As shown in Figure 1-4, the stator
consists of three coils, where the center coil (CL3) is split into two independent coils
(CL31 and CL32), and each of these is connected to one of the outer coils (CL1 and
CL2). In this way, two independent currents (I1 and 12) can be used to control both
a force in the z-direction and a torque about the y-axis. A reluctance actuator can
be subject to parasitic torques that can compromise accuracy. This is especially true
of reluctance actuators with three or more poles. We discuss this phenomenon in further detail in Section 2.6.2. This design therefore provides a method for counteracting
these parasitic torques.
In [67], Ono of Nikon designed a reluctance actuator with a curved target. This
is shown in Figure 1-5. The target is intended to have a stator on either side of it
to make a bi-directional actuator (one stator can generate only an attractive force).
Because the air gap flux is normal to the curved surface, the force generated by
this flux will result in reduced parasitic torques about the target's longitudinal axis
37
Figure 4
ACT1
LG1
LG3
CL1
\
CL3
__
CL32
LG2
CL2
---. _-
I12
DOFI
CL31
DOF2
ACT2
ACT
Figure 5
ACT1
z
Y
X
CL2LG
CL3LG
ACT2
LGl
CL1
G
Figure 1-4: Schematic diagram of ASML's 2-DoF actuator design invented by Hol
showing the coil wiring (top) and 3-D view of the actuator (bottom). Figure is taken
from US Patent 8,472,010 [41].
that arise from misalignments or relative motion between the stator and target. One
disadvantage of this design, however, is that the actuator will not be as efficient in
generating force because the effective air gap is larger and because the flux normal
to the target surface is not perfectly parallel to the driving direction.
R
Figure 1-5: Schematic diagram of Nikon's reluctance actuator design invented by Ono
showing curved target. Figure is taken from US Patent 6,906,334 [67].
Yuan of Nikon developed a way to drive two reluctance actuators as a single unit
in order to simplify the control [82]. Figure 1-6 shows a precision stage (120) driven
38
by reluctance actuators (121 and 122). Reluctance actuators are typically operated
with some bias force FO to avoid the theoretically infinite amplifier slew rates that
would be required at zero force. We discuss this in Chapter 2. The net force F on
the stage is F2 - F1 . Then, to generate a force F to the right, the force F2 generated
by the right actuator (122) is increased to Fo + F/2 and the force F generated by
the left actuator (121) is decreased to Fo - F/2, where F < 2Fo. If the actuators are
operated in tandem in this way, the maximum force achievable is F = 2FO because
reluctance actuators can only generate attractive forces.
Thus, to generate large
However, with a large bias force the average power
forces, FO must be increased.
dissipation increases significantly. Yuan circumvents this problem by using a small
bias force during the exposure (constant velocity) phase of the scan profile when the
net force required is small, but then increasing the bias force during the acceleration
phases when the required forces are large. In this way the large bias force is present
during only portions of the scan, and he is able to operate the actuator pair as a
single unit while minimizing power dissipation.
122
121
X1
Sns1
Sns2
148
147
I
120
141
142
145
F1
143
144
F2
146
Figure 1-6: Schematic diagram of precision stage actuated with two reluctance actuators as a single unit. Figure is taken from US Patent 6,130,517 [82].
1.2.2
Control
Control of a reluctance actuator for force-linearization can be subdivided into two
categories: model-based feedforward control and feedback control. In model-based
39
feedforward control, the desired force is used as the input to an inverse model of
the actuator force. The inverse model then outputs a drive signal to the actuator
(a current, for example). If the inverse model is accurate, the actuator will output
a force that matches the desired force. In feedback control, a signal such as the air
gap flux is sensed and controlled with a high-bandwidth control loop.
These two
categories are not mutually exclusive and can be used together.
Butler of ASML suggests in [19] a control scheme that uses both reluctance actuators and Lorentz actuators. The reluctance actuators are used for the feedforward
force and the Lorentz actuators are used for the feedback force. Since the feedback
force dominates during the constant-velocity exposure phase of the scan profile, the
Lorentz actuators generate the actuation when the position accuracy is more critical.
The feedforward force dominates during the acceleration phases of the scan, so the
reluctance actuators provide a more efficient way of transmitting the force during the
large-force portions of the scan. The idea then is that this scheme reduces the need
for the reluctance actuator to be highly accurate. However, since the position control
bandwidth of the stage is limited, the feedforward force accuracy must still be very
good to meet the position accuracy requirement at the start of exposure. Another
downside of this method is that the volume needed to accommodate both reluctance
actuators and Lorentz actuators is limited.
During a summer internship I had at ASML in 2008, Steve Roux, Mike Carter,
and I conceived a reluctance actuator with force feedback control.
We filed for a
patent application, which was published in 2013 [75]. In this design, a force sensing
element is connected to the stator or target so that the actuator force can be directly
controlled with a high-bandwidth force control loop. With an ideal force sensor, this
is an attractive solution since force is the variable we are ultimately trying to control.
However, real force sensors are problematic because they cannot distinguish between
applied force and inertial forces, so stage accelerations will introduce errors into the
force sensing. Another concept introduced in this patent is the idea of servoing the
fine-positioning stage to different actuator air gaps during different phases of the scan
profile. For example, in the schematic of Figure 1-7, a fine-positioning stage (3) is
40
actuated in the scan axis (x) by reluctance actuators (6 and 7). While accelerating in
the +x-direction, actuator 7 is generating the force. During this acceleration phase,
the actuator 7 air gap is made smaller and the air gap 6 is made larger.
While
accelerating in the -x-direction, actuator 6 is generating the force. During this phase,
the actuator 6 air gap is made smaller and the air gap 7 is made larger. In this way,
we can make better use of the improved actuator efficiency at small air gaps without
necessarily having to tighten assembly tolerances.
Figure 1-7: Schematic diagram of short-stroke stage actuated by two reluctance actautors. Figure is taken from US Patent 8,553,205 [75].
Hsin of Nikon describes a force feedback controller in [43] that utilizes an adaptive
gain adjustment so that the loop gain remains relatively constant even during large
gap disturbances. This allows for improved performance and stability over a large
range of gaps.
On a prototype actuator, a consistent loop crossover frequency of
-60 Hz is demonstrated over a range of 800 lim. In contrast, without the adaptive
gain adjustment, the loop crossover frequency shifts between 25-60 Hz depending on
the gap, leading to deteriorated performance.
In [49], Katalenid uses a Hall sensor and sense coil for flux control using state
feedback.
A sense coil wound around one of the pole faces measures dA/dt and is
used for a high-bandwidth inner loop, where A is the total flux linked by the coil. A
Hall sensor placed on one of the pole faces measures the flux density and is used for
a low-bandwidth outer loop. Combining the sense coil and Hall sensor in this way
provides for innovative way to take advantage of the benefits each sensor offers: the
superior noise performance of the sense coil at high frequencies is realized, while the
41
Hall sensor compensates for the sense coil's inability to measure DC flux. With this
method, a maximum force error less than 0.1 N was demonstrated for a 200 N force
pulse on a 1-DoF testbed with a constant air gap. The main downside of this method
is that the Hall sensor takes up volume in the air gap, which limits how small the air
gap can be made, and thus limits the actuator efficiency.
Katalenid also describes a method that uses an inverse hysteresis model to control
the actuator force via feedforward. The desired force and air gap measurement are
first sent to a lookup table (LUT), which outputs an intermediate variable X. This
intermediate variable is sent to an inverse hysteresis operator, which then outputs
a commanded current. A high-bandwidth current loop then controls the actuator
current to track the commanded current. If the inversion is accurate, the resulting
current will produce the desired force. The variable X does not have any physical
significance and is chosen based on constraints set by the form of the inverse hysteresis
model. This method performs well at a constant air gap for a slowly varying desired
force, exhibiting a maximum force error of less than 0.1 N for a force profile with a
240 N maximum force. The author cautions that because of the hysteresis model's
simplicity, it is limited in its local memory, which limits this method's accuracy for
more complex inputs, such as those resulting from a gap disturbance. Moreover, for
this method to be sufficiently accurate for lithography applications, the air gap must
be known to better than 250 nm.
There are a number of patents in the literature that develop inverse models for
feedforward control.
In [83], a standard model for a reluctance actuator current-
force relationship is inverted to improve the linearity. In [22], additional parameters
are added to the inverse model and then fine-tuned during calibration to generate a
better fit. In [40], an automatic calibration algorithm uses an adaptive controller to
fine-tune the inverse model that can also account for angular misalignments. These
examples also make use of feedback linearization, whereby the gap measurement is
fedback into the inverse model so that the gap dependency can be accounted for in
the model. One thing these methods lack, however, is that they do not model the
actuator saturation or hysteresis. This limits the achievable accuracy.
42
1.3
Thesis Overview
In this thesis, we focus on modeling and controlling the reluctance actuator for forcelinearization. After considering various alternatives for controlling the actuator, we
found flux control to be the best solution for our application. We focus on flux control
using a sense coil for the feedback measurement because of its simplicity, its superior
noise characteristics, and because it can be used without affecting the operating air
gap, unlike other flux sensor alternatives.
Because a sense coil is AC-coupled to
the actuator flux, however, we cannot use it for control at DC. To generate a flux
estimate for control at DC, we develop an actuator real-time hysteresis model based
on the current and gap measurements, which are measurements already available in
the lithography scanner. By combining this model with the sense coil measurement,
we are able to design a flux controller for force-linearization of the actuator. With
this flux controller, we test the force accuracy of the reluctance actuator on a 1-DoF
air bearing stage and experimentally demonstrate a stiffness well below 0.04 N/pim,
which is equivalent to the specification of 0.0001Fmax N/pm for our prototype.
1.3.1
Reluctance Actuator Design
In Chapter 2, we present a detailed design and analysis overview of a reluctance
actuator. A 2-D illustration of a three-pole reluctance actuator showing the primary
flux path for one half of the actuator is shown in Figure 1-8. By producing a reluctance
force normal to the actuator pole faces and by operating at small air gaps, a reluctance
actuator is able to generate much higher force densities (force per unit mover mass)
than a Lorentz actuator, allowing a reluctance-actuator-driven stage to achieve higher
accelerations and servo bandwidths. The primary reluctance actuator nonlinearities
that make accurate force control a challenge are also presented in Chapter 2. We
address potential actuator configurations and control topologies that can be used to
linearize the reluctance actuator.
43
Target
-..
Staor
Flux path
Coil windings
Figure 1-8: Three-pole reluctance actuator.
1.3.2
Reluctance Actuator Modeling
A key contribution of this thesis is the development of an accurate nonlinear model of
the reluctance actuator that can be used for real-time force control. This model must
be able accurately to capture the ferromagnetic hysteresis, saturation, and gap dependency of the reluctance actuator. In Chapter 3, we present the Preisach hysteresis
model, which we use as the building block for our actuator model. We present two
alternative ways to implement the Preisach model. We also present an alternative
hysteresis model, the Chua model, which provides a simpler but less accurate way
to model ferromagnetic hysteresis. In Chapter 4 and Chapter 5, we incorporate the
Preisach hysteresis model into a lumped-parameter reluctance actuator model that
is based on the analysis that was presented in Chapter 2. Using simulations of this
model, we investigate the effects of hysteresis, gap disturbances, and eddy currents
on force accuracy. We also develop theory for predicting errors from these sources.
We develop a model that can be used for real-time flux estimation in Chapter 6.
We present a novel way of modeling the actuator using a 'sheared' model of the actuator hysteresis, whereby we incorporate the linearizing effect of the air gap directly
into the hysteresis model. We term this model the sheared hysteresis model (SHM).
44
We then integrate this model into an observer that treats the changing air gap as a
disturbance, which allows us to make use of a single-variable (one input) hysteresis
model to estimate the flux. We simulate the SHM with observer and show that it is
numerically stable in the presence of dynamic gap changes and that it captures the
actuator behavior into saturation.
In Chapter 7, we extend the actuator lumped parameter model with additional
elements using magnetic circuit techniques and flux tube methods to model the air
gap fringing fields. This augmented model can be combined with the hysteresis model
if additional accuracy is desired.
1.3.3
Reluctance Actuator and 1-DoF Testbed
A photograph of the prototype reluctance actuator stator is shown in Figure 1-9. A
CAD model of both the stator and target is shown in Figure 1-10.
The actuator
stator consists of a ferromagnetic core comprised of two nickel-iron (NiFe) cut-cores
adjacent to each other and a coil wound around the center pole. The target is a NiFe
I-bar. A shielded sense coil is wound around the center pole face of the core for flux
sensing. In the CAD models, this is shown as a printed circuit board (PCB), which
is another possible way to implement a sense coil. In the photograph, the stator is
shown mounted on load cells so that we can measure the actuator force output.
Figure 1-11 is a photograph of the 1-DoF air bearing testbed. The motion stage
uses a linear air bearing with position measurement and feedback via a high-resolution
linear encoder. The linear encoder is also used to provide an estimate of the reluctance
actuator air gap. On one end of the motion stage is the reluctance actuator. On the
other end of the motion stage is a voice coil actuator, used for nulling the reluctance
actuator force and for applying position disturbances to the reluctance actuator.
Using this motion stage, we test our reluctance actuator flux controller and use the
load cells to measure the force accuracy. The detailed design of the reluctance actuator
prototype and motion stage is described in Chapter 8.
45
Support Block
Load cells
Stator core
Stator housing
Sense coil
Figure 1-9: Reluctance actuator stator prototype.
PCB
C-Core
Coil
Coil
C-Core
I-Core
PCB
Figure 1-10: CAD model of reluctance actuator.
46
4
Figure 1-11: Photograph of 1-DoF testbed.
1.3.4
Loop-Widening Investigation
Chapter 9 discusses a loop-widening phenomenon we discovered on the reluctance
actuator prototype: the actuator flux as measured by the sense coil shows a phase lag
with respect to the measured actuator current that increases with frequency. This
loop widening manifests itself at frequencies below 10 Hz, which is too low for eddy
currents to account for it. Figure 1-12 demonstrates the loop-widening behavior by
plotting the measured actuator flux density (B) against the measured current (I)
for different frequencies.
This makes accurate force control via flux feedback more
difficult because the phase lag in the measured flux does not appear in the measured
force. At the time of writing this thesis, our investigation into the cause of the loop
widening remains inconclusive, and we therefore recommend further investigation.
1.3.5
Experimental Results
Finally, we present experimental results for the SHM and the flux controller on the
1-DoF testbed in Chapter 10. We demonstrate good agreement between the real-time
SHM observer flux estimate and the sense coil flux estimate. Figure 1-13 shows the
SHM estimate compared with the sense coil estimate when the actuator is driven
with a voltage sine wave at a nominal air gap of 530 pim and a 35 lim peak-to-peak
47
B
1.5
v. I
B v.I
0.01
-0.5
-1
1
E
0.005
0.5
E
0
C)
Z'
(D
F
-5
Hz
Hz
2 Hz
Hz
0
X0
-0.5
-0.005
'
-1.5
3
'
-2
-1
'
'
'
-1
0
1
2
-0.01
-0. 1
-0.05
0.05
0
current [A]
0.1
current [A]
Figure 1-12: LEFT: B-I data measured on reluctance actuator demonstrating loopwidening phenomenon; RIGHT: zoomed in near origin.
dynamic gap disturbance is applied.
B v. 1
I-
-
sense coil inte r-ted
hysteresis estimate
1.3
I
S1.2
,
-
0.5 F
2 .6
2.7
2.8
2.9
3
3.1
3.2
0.05
-0.5
E
-1
-1.5
-4
-2
0
I [A]
2
0
-0.O5L-0.5
4
0
I [A)
0.5
Figure 1-13: LEFT: SHM observer real-time flux estimate compared to sense coil
measurement with 35 pm p-p gap disturbance about nominal gap of 530 Jm; TOP
RIGHT: zoomed in at saturation; BOTTOM RIGHT: zoomed in at origin.
Another key contribution of this thesis is designing a flux controller for the reluctance actuator that utilizes a sense coil for the feedback measurement. A sense coil
allows us to measure the actuator flux without compromising the air gap. Because a
sense coil measures the flux rate of change, the resulting signal must be integrated to
obtain a flux measurement. The upshot is that the sense coil cannot measure at true
DC. We have therefore designed a hybrid flux estimation scheme that combines the
sense coil measurement with the SHM estimate that is valid at DC. The hybrid estimation scheme uses a complementary filter pair structure such that the flux estimate
48
relies on the sense coil for high frequencies and relies on the SHM at low frequencies.
Using this hybrid estimation scheme, we achieve a 4 kHz crossover frequency for the
flux controller on the reluctance actuator prototype. Combining the flux feedback
with force-flux-gap model inversion control, we demonstrate a maximum stiffness of
0.012 N/pm at a bias force of 35 N for frequencies up to 100 Hz, well below the maximum allowable stiffness of 0.04 N/pm. The measured stiffness frequency responses
for nominal forces of 5 N and 35 N are shown in Figure 1-14.
E:100
-Max allowable
stiffness
-
F(s)/g(s)
E
10-5
10 110
10'
103
2
I AAA
FO = 5N_
F -=35N
0
(D
-1000
C,,
-2000
CL
-3000
-4000
10 1
103
102
104
frequency [Hz
Figure 1-14: Measured frequency responses of flux-controlled reluctance actuator stiffness.
49
50
Chapter 2
Reluctance Actuator Design
In this chapter, we present a general design overview of reluctance actuators for highprecision applications. First, we develop the basic electromagnetic theory behind the
reluctance actuator operation. We also present the electromagnetic theory for Lorentz
actuator operation. We then contrast the reluctance actuator with the Lorentz actuator and demonstrate how a reluctance actuator can achieve much higher force
densities for the range of operating gaps typical in the industry.
Next, we give an overview of the different core materials that can be used for a
reluctance actuator and the trade-off between hysteresis and saturation flux density
that typically exists among these materials. We then discuss the primary nonlinearities of a reluctance actuator that present challenges to high-precision control, chief
among these being nonlinear stiffness, magnetic hysteresis and saturation, eddy currents, fringing fields, and the nonlinear force-flux relationship. We consider how to
linearize a reluctance actuator through actuator configuration, i.e., through geometric
or physical modifications to the basic actuator structure or by driving the actuator in
different ways. We then consider how to linearize a reluctance actuator through various combinations of feedforward control and feedback control, the advantages and
disadvantages of each controller topology, and the different types of sensors available for feedback control. We also discuss other controller methods such as feedback
linearization and iterative learning control. Finally, we discuss alternate reluctance
actuator configurations beyond the basic 'U'-core or 'E'-core configuration.
51
These
include a double-sided reluctance actuator that is capable of generating bi-directional
force, multi-Degree-of-Freedom actuators that can generate forces in multiple degrees
of freedom (DoF), and multi-pole actuators that can generate the same force with
lower moving mass.
2.1
Electromagnetic Fundamentals
In this section, we analyze the electromagnetic theory of the reluctance actuator and
Lorentz actuator from first principles. We then compare the force generated by the
two classes of actuators and show that the reluctance actuator is capable of generating
much higher force densities (force per unit mass) than the Lorentz actuator.
2.1.1
Reluctance Actuator Electromagnetics
The reluctance actuator can be analyzed using Maxwell's equations as presented for
example in [39].
Figure 2-1 is an illustration of a three-pole reluctance actuator
showing the primary magnetic flux path for one half of the actuator. Here, NI is the
number of amp-turns in the actuator coil, g is the length of the air gap, Hg, and
Hg 2
are the magnetic field intensities in the two air gaps of the primary flux path, and
HFe is the magnetic field intensity in the ferromagnetic core and target (also called
the mover). In this analysis we assume symmetry, i.e., the air gap is uniform. We
also assume uniform flux density in the steel, which is of uniform cross-section. We
also analyze the actuator as a 2-D structure with depth d into the paper.
Air Gap Magnetic Flux Density
We first solve for the magnetic flux density in the air gap. This flux is found by
applying Ampere's Law, Gauss's Law, and the material constitutive relationships.
Ampere's law in integral form is
H - dl =
52
J - dS.
(2.1)
H 1
g
1H
Coil windings
Flux path
Figure 2-1: Flux path for a three-pole reluctance actuator
This states that the line integral of the magnetic field intensity (H) around a closed
contour is equal to the surface integral of the current density (J) passing through the
contour. For the flux path being considered in our reluctance actuator example, this
simplifies to
HFeIFe + Hgj9 + H9 g2 = NI.
(2.2)
The variable 1Fe is the length of that portion of the flux path in the core and target.
In this analysis, Hg1 and Hg2 are approximated as being constant in the gap. This
is a reasonable assumption if the gap is small relative to leakage paths and if it is
small relative to the pole-face dimensions. These conditions will obtain if the actuator
is designed properly. Similarly, HFe may have local variations, but for a first-order
analysis we consider its average value in order to represent the ferromagnetic portion
of flux path as a single lumped parameter.
Gauss's Law is
B - dS = 0.
(2.3)
This states that the closed surface integral of the magnetic flux density B must always
be zero. In other words, the flux entering a volume must be equal to the flux that
exits.
53
If the actuator air gap is small relative to the pole face dimensions, the amount
of fringing flux will be small and so the flux density can be assumed to be uniform
within the air gap. This assumption in conjunction with (2.3) can be applied to the
reluctance actuator at the center pole face air gap as
BcAc = B91 Agi.
(2.4)
Equation (2.4) is illustrated in Figure 2-2. The dotted box represents the volume to
which Gauss's Law is applied. Entering one part of the volume through the pole face
area A, is the core flux density B,. Exiting the volume on the target side is the air
gap flux density Bg1 through air gap area Agl.
Figure 2-2: Gauss's Law applied to the center pole face air gap.
Since we have stipulated that fringing fields in the air gap are small, the flux that
enters the volume through area A, will leave through a nearly identical area, i.e.,
Ag, ~ A,. Equation (2.3) then reduces to Bg1 = B,.
A similar analysis can be applied to the left-side air gap in the flux path. If the
left and right pole faces are sized appropriately to be half the area of the center pole
face, then the flux density in the left air gap will be equal to the flux density in the
.
center air gap, that is, B92 = B91 -A B
9
The next step is to relate the magnetic field intensity to the magnetic flux density.
54
The flux density in the air gap is related to the field intensity through the permeability
of free space, bto. The flux density in the core and target material is related to the
field intensity in a nonlinear manner, but for a first-order analysis, this relationship
can be approximated as linear by assuming a constant material permeability P. These
constitutive relationships are
B9 = po Hg,
(2.5)
Be = PHFe-
(2.6)
These relationships can be substituted into (2.2). Recalling that B, = B9
B, this
results in
B
B
-lFe + -2g
p
AO
= NI.
(2.7)
The core permeability, p, is typically at least a thousand times greater than yo when
operating away from the saturation region of the material. If the reluctance actuator
is designed correctly, the first term in (2.7) will be much smaller than the second term
and can be ignored for the purposes of a first-order approximation (i.e.,
2
>>
e).
If we solve for B we deduce
B=
2g
.ONI
(2.8)
Thus, for a first-order approximation, the air gap magnetic flux density is proportional
to current and inversely proportional to the gap.
If we desire a more accurate approximation for the gap flux density, we can include
the core permeability in our calculation. Instead of ignoring the first term in ( 2.7),
we can replace p by prpo, where p, is the relative permeability of the core material.
If we then solve again for B, we arrive at
B=
1
AlFe
As /,
+ 2g
approaches infinity, (2.9) reduces to (2.8).
(2.9)
The relationship can be further
refined by replacing the constant Pr by a single-valued function of H or B, e.g.
pr = p, (B), to account for the nonlinear permeability. Use of such a function permits
55
(2.9) to take into account saturation; however, it also requires iteration to solve. The
nonlinear permeability function does not take into account hysteresis, fringing fields,
or leakage flux. It also does not take into account any local field variations within
the core or target material.
Reluctance Actuator Force
The force on the target can be derived from Maxwell's Stress Tensor' for example
following the approach given in [90]. A magnetoquasistatic (MQS) field produces a
stress tensor with components Ti, on a surface. The subscript i denotes the direction
of the stress and the subscript
j
denotes the direction of the surface normal vector.
Ignoring magnetostriction, this stress is given by
Ttj = pH H,
-
(2.10)
I63pH2,
2
where Hi and Hj are the magnetic field intensities in the i- and j-directions, respectively, and 65j is the Kronecker delta, defined as 0 when i
# j and 1 when i = j, and
H is the magnitude of the magnetic field. In Cartesian coordinates, the force in the
i-direction on a body enclosed by a surface S is expressed as
F =
(x+Tiy + Tiz) dS.
(2.11)
The key to using the Maxwell Stress Tensor effectively is to choose an appropriate
surface over which to integrate. In the case of the reluctance actuator being discussed,
we choose a surface shown by the dotted lines in Figure 2-3.
The flux passing through surfaces 2, 3, and 4 will be zero since we have ignored
fringing and leakage flux in this analysis (if considering three dimensions, this is
also true of the front and back surfaces), so these surfaces will not contribute to the
integral in (2.11). This leaves surface 1 as the only surface contributing to the force
evaluation. Along this surface, the flux density is normal to the surface, and thus the
'Other methods, such as energy methods, can also be used to derive the force
56
3
22
4'
Figure 2-3: Maxwell surface for calculating the magnetic force on the mover.
force from the Maxwell stress tensor reduces to
F =
11
B dS.
2po Js
(2.12)
In other words, the force generated from the flux passing through surface 1 is the
square of the flux density normal to that surface (Bn) integrated over the surface.
In our simplified model, flux only passes through surface 1 at locations directly
above the stator pole faces.
Since the flux density can be approximated as being
uniform at these locations, (2.12) simplifies to
F =
1
2po
(B2 AL + B2 Ac + B2 AR) =
B2 A
.
(2.13)
2po
Here AL, AC, and AR are the left pole face area, center pole face area, and right pole
face area, respectively. The total pole face area is denoted by A, which is the sum of
AL, AC, and ARSubstituting (2.8) into (2.13), we arrive at
F =
8g2
57
.
(2.14)
The force is proportional to the total pole face area and is proportional to the square
of the current. It is inversely proportional to the square of the air gap.
This approximation of the force serves as a useful guideline for a first iteration
in the reluctance actuator design process. Because this analysis neglects many nonlinearities and non-idealities, such as flux saturation, hysteresis, fringing flux, and
leakage flux, it should not be counted on to obtain high-accuracy predictions.
2.1.2
Lorentz Actuator Electromagnet ics
The electromagnetic force on a single particle in free space with charge q moving at
velocity v in the presence of an electric field E and a magnetic flux density B is [39]
F = q(E + v x B).
This force is known as the Lorentz force.
(2.15)
It can be used to derive the force on a
current-carrying wire of length 1 immersed in a field with flux density B [91] (see
Figure 2-4) as
F = Il x B,
(2.16)
where I is the current carried by the wire and we assume charge neutrality in the
conductor. This result can be applied to a group of N conductors immersed in the
same magnetic flux density and each carrying the same current in the same direction.
The total force on the conductors is
F = NIl x B.
A Lorentz actuator operates on this principle.
magnetic field generated by permanent magnets.
(2.17)
A coil is placed in a uniform
When the coil is energized, the
current interacts with the magnetic field to produce a force on the coil. An equal and
opposite reaction force is generated on the permanent magnet structure.
The cross-section of one possible configuration of a Lorentz actuator is shown in
Figure 2-5. A positive current generates a force on the coil in the positive y-direction.
58
B
F
r
Figure 2-4: The Lorentz force on a current-carrying conductor immersed in an external magnetic flux density B.
In a typical application, the permanent magnet structure is attached to the moving
stage, so the force on the stage is therefore in the negative y-direction. The motor
constant of the actuator can be derived from (2.17) as K
= 2NlBg.
Here N is the
number of turns in the coil and 1 is the length of the coil (in the x-direction) within
the magnetic field.
Both sides of the coil interact with the magnetic field, which
accounts for the factor of two. Bg is the magnetic flux density in the air gap between
the permanent magnets.
Flux path
Figure 2-5: Cross-section of a Lorentz actuator.
As was done with the reluctance actuator analysis, we can apply Ampere's Law
59
(2.1) and Gauss's Law (2.3) to the Lorentz actuator to derive the flux density in the
air gap. If we apply Ampere's Law to the flux path shown in Figure 2-6, we get
2Hmt +
2
Hgg = 0,
(2.18)
where Hm is the magnetic field intensity in the permanent magnets, Hg is the field
intensity in the air gaps between the permanent magnets, and t and g are twice
the height of one permanent magnet and the height of the air gap, respectively. The
magnetic field intensity in the back iron is assumed to be negligible because of its high
permeability. The permanent magnets are assumed to be identical to one another and
any additional magnetic field generated by the self-inductance of the coil is assumed
to be negligible (valid for a first-order analysis in which we ignore inductance). The
magnetic field intensities are assumed to be uniform within the permanent magnets
and within the air gap. The Ampere loop is chosen to intersect the center of either
side of the coil. This way the Ampere loop includes a number of amp-turns in the +xdirection identical to the number of amp-turns in the -x-direction and the right-hand
side of Ampere's law is zero.
Flux path
2
t
Figure 2-6: Ampere's law applied to the Lorentz actuator.
We then apply the constitutive laws relating the air gap and magnet flux densities
to their respective field intensities. In the case of the air gap, we use the relationship
60
in (2.5). In the case of the permanent magnet, we have [39]
Bm = po (Hm + M),
(2.19)
where M is the magnetization of the magnet, which is assumed z-directed. Substi-
tuting (2.5) and (2.19) into (2.17), we obtain
2
-_M
t + 2Bg
pto
=
0,
2Bmt + 2Bg
=
2poMt,
2Bmt + 2Bgg
=
2Bt.
Po
(2.20)
In the last line, poM has been replaced with Br, the remanence flux density of the
permanent magnet.
By the same reasoning employed in analyzing the reluctance actuator magnetics,
we find that Bg = Bm via Gauss's Law, i.e., the flux density in the air gap (Bg) is equal
to the flux density in the permanent magnet (Bm). This assumes that fringing flux
and leakage flux from the magnets are negligible. This is a very crude assumption for
the given magnetic structure, but leads to a simple result for understanding actuator
scaling. We can now solve for B9 as
B
=
t
t+g
Br.
(2.21)
Here Bg is equal to the remanence flux density of the permanent magnet multiplied
by an attenuation factor of t/(t + g), the ratio of total magnet thickness (t) to total
magnet thickness plus air gap (t + g).
From (2.17), we can solve for the Lorentz actuator force in terms of the permanent
61
magnet remanence and given actuator parameters as
F = ()2NlBI.
(t+g
(2.22)
The motor constant is thus K = -LNlB,.
t+g
2.1.3
Comparison between Reluctance Actuator and Lorentz
Actuator
In this section we compare the normal stress generated by the reluctance actuator
to the shear stress generated by the Lorentz actuator. The stresses generated by the
two actuators can be understood by referring to Figure 2-7.
Lorentz Actuator
Reluctance Actuator
V2t
gL
/2t
gR
Hi, B,
Kwr
\
i
" B.
T = 2HBj
H = 0.5Kk= 0.5N1/w
Bj = p.A~t/(gL,
T,= 0.5HBi
Hi = NI/gR
)
Bi = pOHj
Figure 2-7: Force density comparison between a Lorentz actuator (left) and a reluctance actuator (right)
In the left illustration T 3 is the expression for the stress on the Lorentz actuator
coil as derived from the Maxwell stress tensor (see (2.10)). The stresses on both the
62
upper and lower surface of the coil are identical, so they have been combined, giving
rise to the factor of two. The relevant magnetic field intensity (Hi) and magnetic flux
density (Bj) that produce the stress are orthogonal, creating a shear stress on the
coil. If the coil current is approximated as a sheet current denoted by Kk, then Hi on
either side of the coil has a magnitude equal to 0.5Kk. This result can be derived from
applying Ampere's Law to an interface with a sheet current [91]. The sheet current
K
is equal to the number of amp-turns (NI) in the coil divided by the coil width,
w. The magnetic flux density produced by the permanent magnets (shown in blue
and red, where blue represents the magnet north pole and red represents the magnet
south pole) was derived in (2.21) and is proportional to the ratio of the magnet height
to the magnet height plus air gap (t/(t + gL))
.
It is also proportional to the magnet
remanence ([toM). We write Tij for the Lorentz actuator as
t
AL+ t
pOMNI
W
=
t )
g9L + t
oMJAw
W
(2.23)
-
Ti
=
In the second line, we have replaced NI with JAw, where J is the coil current density
If the gap between the coil and the permanent
and Aw is the coil window area.
magnets is small relative to gL, we can approximate Aw as gLw. Then (2.23) becomes
(t
Tij,L
=
\gL
+ t/
poM JgL.
(2.24)
In the right illustration Ti% denotes the Maxwell stress for the reluctance actuator.
In contrast to the Lorentz actuator, the stress is a normal stress rather than a shear
stress. The magnetic field intensity (Hi) in the air gap is proportional to the ampturns and inversely proportional to the air gap
(gR).
This can be derived from (2.2) by
noting that HFe is approximately zero. The magnetic flux density Bi is proportional
63
to Hi. We can write Ti for the reluctance actuator as
2 2
poN I
2g
'
"ii
'0 J2A 2
2g
Ti
2gR2
.
(2.25)
If we compare (2.25) with (2.24), we can note several things. First, the stress produced
by the reluctance actuator increases quadratically with J, while the stress produced
by the Lorentz actuator increases only linearly with J. At high current densities then,
we can generate higher force slew rates (change of force with respect to current) with
a reluctance actuator than we can with a Lorentz actuator. Second, by reducing the
reluctance actuator air gap, we can increase the stress generated by the reluctance
actuator, since T. is inversely proportional to gR. This dimension is limited only by
assembly and crash tolerances in a typical application, so can be made quite small
(on the order of 1 mm). Third, for the reluctance actuator, the motion is in the same
axis as gR, so Tj will change with displacement, i.e., it will have a non-zero stiffness.
In contrast, the motion of the Lorentz actuator is orthogonal to gL, so the Lorentz
actuator will have zero stiffness based on this analysis.
Thermal constraints limit the maximum current density that can be achieved in
a Lorentz or reluctance actuator. For a Lorentz actuator, in order to increase the TIj
for a given maximum current density, we must increase t or gL. However, increasing
t will increase the moving mass. We can instead increase gL, but if the ratio of gL/t
becomes too large, the analysis from Section 2.1.2 will not remain valid, as the flux
from the magnets will no longer couple with the coil.
For a reluctance actuator,
(2.25) indicates that once we have reached the thermal constraints that limit J, we
can simply increase A, in order to increase Tj. Eventually however, we will be limited
by the saturation flux density of the material. From (2.13), if we set B = B8, where
B, is the saturation flux density, and note that Ta = F/A, then we can write the
maximum Tu as
max(T2 )
=
64
2B 2
Ao
.
(2.26)
Thus, for a reluctance actuator, the saturation flux density sets a hard limit to the
achievable Ti.
2.2
Core Materials
To design a reluctance actuator properly, an appropriate ferromagnetic material must
be selected for the reluctance actuator core and target.
The key parameters for
selecting a core material are saturation flux density (B,), coercivity (Hc), permeability
(p), and electrical conductivity (-).
Saturation flux density is the maximum magnetic flux density that the material
can sustain. Higher saturation flux density permits higher maximum force capability for the same actuator size, or alternatively, lower actuator mass for the same
maximum force capability.
Coercivity is a measure of how wide the hysteresis loop is. It is defined as the
magnitude of magnetic field necessary to drive the flux density of the material to zero
from saturation [15]. Alternatively, it is sometimes defined as the field necessary to
drive the magnetization to zero from saturation. For soft magnetic materials, these
two definitions result in values that are nearly identical. On the B-H major loop, this
will correspond to the magnitude of H where B is zero. Higher coercivity results in
larger force errors (if the hysteresis is left uncompensated) and increased core losses.
Figure 2-8 indicates B, and H, on the major hysteresis loop. The remanence flux
density, Br, is also shown.
Permeability is the magnetic analog to conductivity in an electrical medium. It is
defined as the ratio of flux density to field strength (p = B/H) [15] and is a nonlinear
function for soft magnetic materials.
Higher permeability leads to lower magnetic
potential drop in the core and target material and thus to a more efficient actuator
(see (2.9)). As such, a higher permeability material will lead to higher force for the
same number of amp-turns and the same air gap.
Electrical conductivity coupled with permeability determine the extent of eddy
currents in the material. Eddy currents increase with increased electrical conductivity
65
and permeability and degrade the actuator's performance at higher frequencies.
B
Descending Curve
H
H,
Ascending Curve
Figure 2-8: The coercivity (He), saturation flux density (B,), and remanence (B,)
are indicated on the major hysteresis loop.
In general, there is a trade-off between a material's saturation flux density and
its coercivity. A material with high saturation flux density will usually have a higher
coercivity, and thus a wider hysteresis loop. Table 2.12 lists the coercivity and saturation flux density values for some typical materials. In the rest of this section, we
take 49% NiFe as our baseline for comparison, because this is the material we used
for the prototype actuators we designed (see Chapter 8).
Table 2.1: Comparison of key parameters for several core materials
Saturation
Material
Coercivity (A/m)
flux density (T)
50% Ni-50% Fe
49% Ni-49% Fe
49% Co-49% Fe
3% Si-97% Fe
9.6
4.0
14.4
31.8
1.6
1.5
2.3
2.0
Nanocrystalline
0.6
1.2
The CoFe material has the highest saturation flux density. Because of the squared
dependency of force with flux, CoFe can achieve a significantly higher maximum force
2
The information on coercivity and saturation flux density values was obtained from Magnetic
Metals [57]
66
than some of the alternatives. For example, a roughly 50% increase in saturation flux
density over 49% NiFe will result in a 130% increase in maximum force capability for
the same pole face area. CoFe therefore has the highest potential for high force-tomass ratio, which makes it an attractive choice for high force density applications.
The next best alternative for achieving high force capability is SiFe. SiFe has a
high saturation flux density that will result in an 80% improvement in force capability
over 49% NiFe, and it provides a lower cost alternative to CoFe. SiFe is used for many
motor applications and can be obtained at a much lower price than CoFe (it is also
less expensive than NiFe).
Nanocrystalline material has the lowest coercivity of all the materials listed. This
will result in much lower errors from hysteresis.
One of the errors associated with
hysteresis is the residual force error, defined as the force offset that results upon
increasing the current to saturation levels and then reducing it back to zero.
In
Chapter 4, we show this error to increase with HC (see (4.13)). As a result of this
squared dependency, Nanocrystalline material will have a residual force error of only
3% of 49% NiFe for actuators of the same size. However, due to saturation limits, it
will only reach 50% of the maximum force capability of 49% NiFe.
2.3
Actuator Nonlinearities
A reluctance actuator is subject to nonlinear behaviors not present in a Lorentz
actuator. These nonlinearites must be compensated for when used in high-precision
applications. In this section we give a brief overview of some of these nonlinearities.
2.3.1
Force-flux Relationship
The nonlinearity most immediately apparent is the squared dependency of force on
flux (2.13). One consequence of this is that a reluctance actuator is a uni-directional
force actuator, i.e., it can generate force in only one direction. This is one downside
of a reluctance actuator compared to a Lorentz actuator, which can generate both
positive and negative forces. Another consequence of the squared dependency is that
67
the slope of the F-B curve is zero at B = 0, so to command a non-zero dF/dt at
B = 0 requires infinite dB/dt, i.e., an infinite slew rate on the actuator drive current.
Thus, reluctance actuators are usually operated with some bias level.
2.3.2
Hysteresis and Saturation
The B-H curve of a ferromagnetic material exhibits hysteresis and saturation. This
is translated into the B-I (flux-current) domain and into the F-I (force-current) domain, where the effect of hysteresis is amplified because of the squared dependency.
See Figure 2-8 for a B-H curve with hysteresis and saturation.
Saturation can be
compensated using a nonlinear lookup table between current and force. Hysteresis is
more difficult to linearize when controlling only the current since it is a complex function of the drive current and past history. Some possibilities for linearizing hysteresis
are discussed in Section 2.5.
2.3.3
Eddy Currents
Faraday's Law tells us that a changing flux will generate eddy currents in an electricallyconductive material. Via Ampere's Law, these eddy currents will in turn generate
a counteracting magnetic flux, degrading actuator performance.
Eddy currents are
challenging to model and contribute to the non-ideal behavior of the actuator. Eddy
currents are not accounted for in (2.2), because there we only considered the actuator
coil current and neglected any potential eddy currents in the ferromagnetic core.
2.3.4
Gap Dependency
From (2.14), a nonlinear dependency on gap is evident. Because of the inverse dependency, a negative stiffness results in the actuator. In contrast, a Lorentz actuator
has zero-stiffness in its driving direction. This is ideal for a system in which the longstroke stage does not perfectly track the short-stroke stage: tracking errors will not
translate to force disturbances on the short-stroke stage. With a reluctance actuator,
tracking errors result in force disturbances if the negative stiffness is not compensated.
68
2.3.5
Fringing Flux
If the air gap flux is controlled directly (e.g., via a Hall-effect sensor), some of the
nonlinearities discussed in the foregoing sections are mitigated.
Since there is no
hysteretic relationship between force and flux, the saturation and hysteresis nonlinearities are linearized by accurate flux control. Likewise, from (2.13), we see there is
no gap dependency between force and flux.
However, owing to fringing fields in the air gap (which our first-order analysis
neglected), the relationship between force and measured flux will have some dependency on gap. As the gap changes, the uniformity of the air gap flux changes (for
example, as the gap increases, the flux will spread out more), and the relationship
between force and measured flux changes.
Equation (2.12) is still true, but it no
longer reduces to (2.13) if the field is not uniform.
2.4
Linearization via Actuator Configuration
For precision applications, it is necessary to develop methods that linearize the actuator. These methods can be divided into two general categories: actuator configuration
and actuator control. This section will address actuator configuration. Actuator configuration can refer to physical or geometric modifications to the actuator itself. It
can also refer to alternate ways of driving the actuator.
2.4.1
Current-Biased Linearization
One way to improve the linearity of a reluctance actuator is to use current-biased
linearization.
In this method, two reluctance actuator stators are placed on either
side of the mover (Figure 2-9).
1 and 12 are the excitation currents in the left and
right stators, respectively, and gi and g 2 are the left and right operating air gaps,
respectively. If the air gaps are equal, then gi
69
= 92
A go, and following an analysis
similar to that in Section 2.1.1, we can solve for the actuating force on the target as
F=
CTIof,
(2.27)
90
where C is a constant that depends on geometry and number of turns, 1 is the bias
current defined as (I, + 12)/2, and I is the current difference between the two stators,
defined as (I1 - 12)/2. For a more detailed analysis, see [56], Section 2.2.
Note that the squared dependency of force on current from (2.14) has been eliminated. If 1 and go are kept constant, we have a force actuator that is linear with
current.
However, for scanning lithography applications, the presence of gap disturbances
means that the constant gap assumption cannot be maintained. As a result, (2.27)
will no longer be accurate. Moreover, hysteresis, saturation, and eddy currents remain
unaddressed. Finally, to make full use of the reluctance actuator force capability in
this configuration, the bias current must be set to a high value. This will result in high
power-dissipation levels even when generating zero net force or when at standstill.
g,
Figure 2-9: A current-biased reluctance actuator.
70
2.4.2
Flux-Biased Linearization with a Permanent Magnet
Another way to improve actuator linearity is by using a permanent magnet to bias
the flux. There are a number of ways to implement such a scheme. One such way
is shown in Figure 2-10 as presented in [56]. The force generated by this actuator is
derived in [56] and is given by
poaNI 29
+ -- B
90g o
.
(2.28)
/
A
o
F = -Bo
Here, A is the area of the mover normal to the direction of motion, Bo is the DC bias
flux generated by the permanent magnet, NI is the total number of amp-turns in the
two stator coils, go is the air gap when the mover is centered, and g is the deviation
of the air gap from this nominal position (i.e., the gap between the target and stator
on the left side is 9L = go + g and the gap on the right side is gR = go - g). The
advantage of this actuator is that the force is now linear with both current and gap.
While the stiffness is still finite and negative in sign owing to the gap dependency,
a linear gap dependency is more manageable than the inverse quadratic dependency
in a standard reluctance actuator topology. Unlike the current-biased linearization
scheme, the linearity of the flux-biased actuator does not depend on the mover being
centered. This approach is also more efficient since the bias flux is generated by a
permanent magnet rather than by a bias current.
One downside of this flux-biased topology is that there is also a negative stiffness
in the direction of permanent magnet magnetization (orthogonal to the motion direction).
In the fast-tool servo application described in [56], this is not a problem
because the mover is constrained by a rubber bearing. In a lithography tool however,
the short-stroke chuck is magnetically levitated in all six degrees of freedom and so
this negative stiffness could cause significant servo errors in the non-scanning axes.
Another related problem is that Bo only remains constant insofar as the gap
between the mover and the permanent magnet remains constant. Again, in [56], this
problem is solved by means of the rubber bearing. In a lithography tool, there is no
such lateral constraint.
71
Coil flux
Direction of motion
Figure 2-10: The flux-biased actuator with a permanent magnet generating the bias
flux. Design presented by Lu [56].
One possible solution to the first problem is to orient the flux-biased actuators such
that the permanent magnet magnetization is in the direction of gravity. This would
allow for the permanent magnets to double as gravity compensators.
Lithography
tools already use permanent magnets in their gravity compensators, so this could be
an effective way to achieve high linearity in the scan axis while using the permanent
magnet stiffness in the vertical axis to its advantage. Nonlinearities such as hysteresis
and saturation still remain because the analysis used to derive (2.28) does not take
these into account.
2.4.3
Flux-Biased Linearization with an Electromagnet
A variation on the permanent magnet flux-biased actuator is to replace the permanent
magnet with an electromagnet that generates the bias flux. See Figure 2-11. The two
coils are driven in common-mode (CM) to generate the bias flux and are driven in
differential-mode (DM) to generate the driving force on the target. The advantage
here is that the DC current could be set to zero during wafer exposure so that the
72
negative stiffness orthogonal to the drive axis would be eliminated during this part
of the scan. However, there are several disadvantages.
The actuator would be less
efficient since a current rather than a permanent magnet is being used to drive the
DC flux. The gap between the DC electromagnet and the mover (x) would have to be
significantly larger than the air gaps in the scanning direction (g and gR) so that AC
flux would not couple into the DC path. Since the gap between the DC electromagnet
and mover would therefore be relatively large, driving the DC coil would result in
larger power losses. A further complexity is managing the 'switching off' period of
the DC electromagnet during exposure, which cannot be done instantaneously owing
to the coil inductance and amplifier slew rate.
Alternatively, the center electromagnet could be kept active during the entire scan
and used for actuating one of the non-scanning axes. This would allow for a compact
2-DoF actuator; however, the linearity of the actuator in the scan direction is lost
and the control becomes more complicated because the force in the scanning axis is
now dependent on both the central coil current and the outer coil currents.
AC flux
Outer coils generate
lw
no,
60-AC
flux
Center coil generates DC flux
Figure 2-11: A flux-biased actuator with an electromagnet generating the bias flux.
73
2.4.4
Operating Regime
Intelligent choice of operating regime can lead to improved linearity for reluctance
actuators.
One way to do this is by operating away from the saturation region.
This results in a more well-behaved actuator, as the actuator will approximately
have the same permeability throughout the entire operating regime, and the F oc I2
relationship will therefore remain approximately valid. The hysteresis nonlinearity
can also be reduced by operating away from saturation.
In high-acceleration applications in which achieving a low moving mass is critical,
operating away from saturation is not an ideal solution because we are not taking full
advantage of the force capabilities of the reluctance actuator: the mover mass will
need to be made larger to generate the required force.
Another way to improve the actuator linearity is by increasing the nominal operating air gap. Consider again (2.7), rewritten here as
B
HFelFe + -2g
po
(2.29)
= NI,
where we have substituted (2.5) for Hgi and Hg 2 . If we denote the applied magnetic
field in the core as Ha = NI/Fe and the ratio of total air gap length to mean
ferromagnetic path length as n = 2g/lFe, then we can write (2.29) as
HFe +
HFe +
/yo
nB (HFe)
p'o
= Ha,
=
(2.30)
Ha.
In the second line we have shown B as a function of
HFe
explicitly. We want to
determine the B-Ha relationship from the material hysteresis curve.
We plot the
B-HFe material hysteresis curve in Figure 2-12, where it is indicated by the blue
curve.
If the air gap were zero, the B-HFe relationship would be identical to the
B-Ha relationship. In this case, Ha = HFe, which can be seen by setting n = 0 in
(2.30). To determine Ha when the air gap is not zero, we take the HFe values from
74
the material hysteresis curve and add the quantity nB/po to it, where B = B(HFe).
The result is shown by the dotted green curve in the figure. The original hysteresis
loop is 'sheared' by the amount nB/po, thereby making the B-Ha relationship more
linear. Note that the larger the ratio n (larger air gap), the greater the amount of
shearing and linearization that occurs.
No air gap
Shearing
line
nB
nB
-rl
P0
With air gap
PO.
H
Ha-HLe
Figure 2-12: An actuator air gap causes the hysteresis loop to shear.
The linearization that results from shearing occurs because the linear air gap
reluctance is large compared to the nonlinear material reluctance, and so the air gap
magnetic field H. dominates over HFe. Therefore, a change in the applied magnetic
field Ha will generate a change in B that has a greater linear component to it when
an air gap is present.
A further advantage to increasing the nominal operating gap is that gap disturbances have less effect on the force output, i.e., the magnitude of the stiffness is
reduced. The downside of increasing the nominal operating gap is that more current
is required to achieve a given force, thus dissipating more power. As can be seen from
Figure 2-7, the normal stress is inversely proportional to the square of the air gap.
Moreover, because a larger air gap will result in a less-uniform flux density in the air
2
gap (i.e., more fringing), the B relationship results in a further lower total force, and
the target will need to be enlarged to achieve the same maximum force.
75
2.4.5
Target Geometry
Various modifications to the target geometry can improve the linearity of the actuator.
Here we address two possible modifications.
One simple way to improve the performance of a reluctance actuator is by oversizing the target area relative to the stator pole face area in the off-drive-axis dimensions
(x- and z-dimensions). An oversized target in the x-direction is shown in the left side
of Figure 2-13.
In this way, the actuator is made less sensitive to off-axis relative
movement between the stator and target. To a first-order approximation, an oversized target will not generate any change in driving force from a relative motion in the
x-axis or z-axis. The penalty paid is increased mover mass. In contrast, a target that
is not oversized will result in changes in driving force when relative motion occurs.
The reason for this is that in the oversized case, the air gap flux in the left and right
air gaps travels through the same target area even when the target moves relative to
the stator, whereas in the non-oversized case, the air gap flux in the left and right
air gaps travel through a target area that varies as the target moves relative to the
stator.
One method to reduce the sensitivity to changing fringing fields as the gap changes
is to design the target with 'teeth' the same size as the stator pole faces. See the
right-side illustration in Figure 2-13. Since fringing fields now have a larger air gap
to traverse, and thus a higher reluctance to overcome, the fringing fields will be
reduced. More of the field will be concentrated in the tooth. However, in contrast
to the oversized-target design, the actuator force will become much more sensitive to
relative off-axis motion between stator and target, and for this reason, this is not a
strong candidate for lithography machines, which require 6-axis motion.
2.4.6
Laminating the Stator Core and Target
In an application that requires high-force accuracy over a broad frequency range, it
is critical to laminate the core and target material. A solid chunk of steel used as
the core or target material will result in internal eddy currents that will degrade the
76
x
Figure 2-13: LEFT: reluctance actuator with oversized target. RIGHT: target with
teeth.
accuracy and performance of the actuator. Moreover, the presence of eddy currents
will increase the core losses.
The standard way to reduce eddy currents is to assemble the stator core and target
with thin strips of permeable lamination material separated by insulating material.
The thinner the lamination, the more effective it will be at breaking up eddy currents.
An important parameter regarding laminations is the stacking factor: the ratio of
permeable material thickness within a stack of laminations to the total stack thickness
(which includes permeable material plus insulating material). As the laminations are
made thinner, a higher proportion of the total thickness is taken up by insulation.
This reduces the effective area for generating force.
min). Magnetic Metals lists the
A typical lamination thickness is 0.004" (~100
stacking factor for this lamination thickness as 90%. For comparison, a thinner lamination of 0.001" (~25 .im) has a stacking factor of only 75-83%, which will result in
a noticeable force reduction for the same area. A key parameter for selecting lamination thickness is the skin depth. The skin depth is a measure of how far an applied
magnetic field will penetrate a ferromagnetic material at a given frequency [39]. It is
defined as the distance from the surface of the conductive material at which the flux
density has decreased to 1/e of its surface value. The skin depth 5 is given by
6= 2 F
,
77
(2.31)
where o, is the electrical conductivity and
f is
the frequency of the applied magnetic
field signal. The skin depth is inversely proportional the square root of frequency.
For the magnetic field fully to penetrate the magnetic material, it is important to
select a lamination thickness that is well below the skin depth for the frequencies of
interest. Skin depth is described in more detail in Chapter 5.
When designing a reluctance actuator with laminations, one consideration to take
into account is how to orient the laminations. One option is to use stamped laminations in the shape of a 'U' or an 'E' and stack them on top of one other until the
desired stator core height is achieved. Figure 2-14 shows an illustration of this. This
is a standard method for building transformers. The target is assembled in a similar
manner, but using a simple rectangular I-shape lamination rather than U-laminations
or E-laminations. One downside of this is that anisotropic material cannot be correctly oriented in all of the flux path.
Figure 2-14: LEFT: a single E-lamination. RIGHT: an E-lamination stack.
Another option is to use tape-wound cut-cores. In this method, a thin strip of the
core material is wound around a bobbin until the desired pole-face width is achieved.
The resulting core is then cut in the center to make a 'U'-shaped actuator core. Two
cut-cores can be placed side-by-side to make a three-pole actuator, see Figure 2-15.
The target is assembled in a similar way, except that instead of cutting the tapewound core in the center to form two U-cores, cuts are made at either end and the
78
resulting middle straight sections are used.
Figure 2-15: TOP LEFT: a tape-wound core with cuts in the center. TOP RIGHT:
a tape-wound cut-core showing the lamination orientation. BOTTOM: two cut-cores
placed side-by-side to form a three-pole actuator stator core.
In a cut-core actuator, the laminations are oriented differently from an actuator
made from a stack of stamped laminations.
In contrast to a U-core stack or an
E-core stack, the grain orientation of the laminations is in the same orientation as
the magnetic flux throughout the entire cut-core actuator.
This results in higher
core permeability (better for efficiency) and a smoother transition into the saturation
region (better for control) [54].
Instead of laminated cores, an alternative way to reduce eddy currents is to use
powder pressed iron cores. A powdered core consists of many small iron particles
with polymer coatings pressed together. Because of the insulation between adjacent
particles, these cores are very effective at breaking up eddy currents. However, the
insulation acts as distributed air gaps within the core, which results in a much higher
effective core reluctance.
The permeability of the core is much lower than with a
79
laminated stack or with a laminated cut-core, resulting in lower efficiency and reduced
force generation.
2.5
Linearization via Control
A reluctance actuator can also be linearized through control. In this section, we
present some basic control strategies using standard feedforward and feedback techniques. We then present an overview of various potential sensors for feedback control.
Finally, we present some advanced control methods that can be used further to enhance performance.
2.5.1
Standard Feedforward and Feedback Control
A combination of model-based feedforward control and local sensor-based feedback
control can be used to linearize the reluctance actuator. The specific controller topology chosen depends on the sensors available for feedback and the required accuracy.
Below we present several different potential controller topologies for a reluctance actuator that use combinations of feedforward control and feedback control. As a baseline,
we assume that the reluctance actuator will have accurate current sensing, as this is
also required for control of Lorentz actuators. Figures 2-16 and 2-17 show the block
diagrams of various controller configurations.
Controller Topology with Current Sensing
Figure 2-16A shows a block diagram of a controller topology for a reluctance actuator in which only the current is sensed.
Here Fd, Bd, and Id are the desired
actuator force, desired flux density, and desired current, respectively. Variables F,
B, and I are the actual force, flux density, and current, respectively. The dotted
blue box shows the current feedback loop. If the current feedback loop is sufficiently
high bandwidth, the blue box can be approximated as a unity gain between Id and
I. In this topology, feedforward is necessary to invert the F(B, g) and B(I, g) relationships. The most basic form of feedforward in this case is to invert (2.8) or (eq:
80
flux'density'w'core'permeability) for I(Bd, go) and (2.13) for B(Fd, go), where go is
the nominal operating gap. A more accurate implementation is to use 1-D lookup
tables based on real data rather than a priori analytic equations for the I(Bd, go)
and B(Fd, go) functions.
Alternatively, one can fit analytic functions to real data
and then use these analytic functions for feedforward. These two latter options allow
for magnetic saturation to be taken into account. Augmenting the I(Bd, go) with an
inverse hysteresis function can also improve the feedforward accuracy.
The dotted green box in the figure shows an additional feedforward path, which
can be used to enhance performance further. Ideally, FF = Pj-1, where P, is the
part of the actuator plant enclosed by the current feedback loop. Here, P, represents
the relationship between actuator drive voltage and actuator current, or, in linearized
form, PI(s) = 1/(Ls + R), where L is the actuator inductance and R is the actuator
resistance.
Two main challenges present themselves with a controller scheme that includes
only a current sensor. The first is that because the B(I, g) and F(B, g) relationships
depend on gap, the controller will suffer from large innacuracies if the changing gap is
not taken into account. The second is that the complexity of the I(Bd, go) relationship
makes accurate modeling a challenge, particularly the modeling of magnetic hysteresis.
When the controller must operate at high bandwidth, the computation time
available for computing the output of a complex hysteretic model is much reduced.
Additional phase lag between current and flux can also arise from eddy currents,
making the I(Bd, g) relationship a dynamic (and nonlinear) one, and thus further
increasing the complexity.
Controller Topology with Current Sensing and Gap Sensing
Given that the operating gap will change dynamically by tens of micrometers during
scanning, a controller that uses only current sensing will likely be unable to meet the
required accuracy specifications. One way to rectify this is by adding a gap measurement. Figure 2-16B shows a block diagram of controller topology for a reluctance
actuator in which the current and gap are both sensed. With the gap known, more
81
A. Controller topology
with cL rrent sensing
Optional
feedforward path
FF,
Fd
B,
I
U
1d
I(Bd 1g0)
SB(Fd,go)
I
B
CP,
B
F(~)F
()-
BIg
Current
controller
------------ --------- ' Actuator
Current
plant
feedback loop
Plant
inversion
B. Controller topology with
current s ensing and gap sensing
Optional
feedforward path
-
Gap measur ment
FF,
d
B(Fd,g)
d l(Bd,g)
C,
1d
{
U
P,
B(Ig)
I
BF(B,g) --
Current
controller
Gap measurement
----------------------Current
feedback loop
Plant
Actuator
plant
inversion
C. Controller topology with
flux sensing
Optional
feedforward path
FFB
Fd
'o
SB(Fd~o
Bd
CB
D. Controller topology with
flux sensing and gap sensing
B(Fd,g)
Plant
inversion
B
' Bj
FBg
Flux feedback
Actuator
loop
plant
F
Optional
feedforward path
Gap
measurementi
Fa|
B
controller
I%
invesion
inversion
u
Flux
I
Plant
+
du
B++
FFB
B |)F
CB
d
PB
Fr
g
Fu
controller
Flux feedback
Actuator
loop
plant
Figure 2-16: Block diagrams for four reluctance actuator control strategies.
82
E. Controller topology with flux
sensing and current sensing
Optional current
feedforward path
Optional flux
feedforward path
FF,
FF
!B
B
Fd-
Iul
P
Cd
C
B(g)
P ntcontroller
nero
I
B(1,g)i F(B,g)F
controller
_-__----_------------' Actuator
Currentpln
feedback loop
Flux feedback
loop
F. Controller topology with
force sensing
Optional
feedforward path
FFF
F
U
Fj++
CF
F
Force
Actuator
controller
plant
Force
feedback loop
G. Controller topology with force
sensing and current sensing
Optional current
feedforward path
Optional force
feedforward path
FF,
FF
Fd I
F
d
Force
-----controller
,
U
P,
IB(1,g)
BF(B,g)
Current
-
IL------------------------------------controller
C--- r--n---Actuator plant
feedback loop
Force
feedback
loop
Figure 2-17: Block diagrams for three additional reluctance actuator control strategies.
83
accurate 2-D lookup tables can be implemented for the I(Bd, g) and B(Fd, g) functions.
An inverse hysteresis model that depends both on desired flux density and
operating gap can also be implemented for improved accuracy.
In a scanning lithography tool, measurements from differential sensors between
the long-stroke stage and short-stroke stage are already available and can be used
to estimate the gap. Since these sensors are not located in the reluctance actuator
gap, it is necessary to transform these measurements into a local actuator coordinate
system.
The price paid for adding a gap measurement is the increased complexity of
the feedforward models.
The lookup tables become 2-D lookup tables instead of
1-D lookup tables, and the inverse hysteresis model likewise becomes multivariabledependent. The feedforward is also subject to errors arising from inaccuracies in the
gap measurement: these include measurement delay, imperfect coordinate transformation, sensor nonlinearity, sensor noise, and the effect of flexible-body dynamics
between the sensor location and the actuator air gap.
Controller Topology with Flux Sensing
If a flux sensor is used, the feedforward model becomes much simpler.
This con-
figuration is depicted in Figure 2-16C. If the flux feedback controller is sufficiently
high-bandwidth, the blue box can be approximated as a unity gain between Bd and
B. If the flux loop is sufficiently accurate and the relationship between force and flux
has minimal dependency on gap, it is now only necessary to implement a B(F, go)
feedforward function. The I(Bd, g) feedforward function and inverse hysteresis function are no longer necessary since Bd is achieved via a high-bandwidth feedback loop
around the flux sensor (shown by the dotted blue box), although these can still be implemented in the optional feedforward path (dotted green box) if further performance
enhancements are desired.
The flux controller will also tend automatically to correct for any flux reduction
arising from eddy currents.
The upshot is that the B(Fd, go) inverse relationship
required can be modeled as a static relationship even if eddy currents are present,
84
and thus will be easier to model accurately than the I(Bd, g) relationship, which may
require a frequency-dependent model in order to compensate for eddy currents.
Because of fringing field effects on force, a gap measurement is likely still needed
for high-accuracy control even with flux feedback. Figure 2-16D is a configuration in
which both the flux and the gap are sensed. However, unlike with the I(Bd, g) relationship, this gap dependency is a second-order effect, so that the gap measurement
accuracy and resolution needed for high-accuracy control with a flux sensor will be
much reduced.
A third flux feedback configuration is shown in Figure 2-17E. In this configuration, the current is sensed as well. This allows us to include an inner high-bandwidth
current feedback loop (dotted red box). If the bandwidth of this inner loop is sufficiently higher than the flux-controller bandwidth, the red box can be approximated
as a unity gain between Id and I from the perspective of the flux controller. This
permits simpler controller design and higher bandwidth of the flux loop. The current
controller will provide additional phase that compensates for the actuator inductance.
In this way, the only phase loss that the flux controller has to compensate for is the
phase loss from hysteresis and the phase loss from eddy currents. The phase loss from
hysteresis will be minimal because of the linearizing effect of the operating gap, and
the phase loss from eddy currents should also be minimal up to very high frequencies
if the actuator is designed properly.
A gap measurement (not shown) can also be
included in this controller topology. Optional feedforward paths for both the flux and
current can be included for improved performance.
Controller Topology with Force Sensing
Another type of controller topology includes a force feedback loop as shown in Figure 2-17F. If the force feedback loop is sufficiently high bandwidth and accurate, no
additional feedforward is necessary.
In terms of controller complexity, this topol-
ogy is the simplest. Note also that no gap measurement is required in this scheme.
The main drawback of this topology is the non-trivial challenge of implementing an
accurate force sensor.
One of the biggest challenges with force sensing is that the
85
acceleration of the actuator introduces an error into the force measurement. Another
challenge is that the force sensor introduces a compliance into the mechanical force
loop, which compromises bandwidth and increases the gap disturbance magnitude.
A third challenge is that since the reluctance actuator can only generate force in one
direction, the controller must be designed so as not to command a signal that results
in a negative flux.
This controller configuration is augmented with a high-bandwidth current control
loop in the topology of Figure 2-17G. This current control loop serves the same
purpose that the current control loop in the flux feedback topology in Figure 2-17E
served: it linearizes the plant as seen by the force controller and thus permits higher
loop bandwidths and simpler controller designs. Other configurations not shown are
also possible. For example, instead of a minor high-bandwidth current loop, we could
include a minor high-bandwidth flux loop. Alternatively, we could include two minor
loops, the innermost loop for the current and a second minor loop for the flux.
2.5.2
Sensors for Feedback Control
One of the key decisions in the design of a reluctance actuator is selecting an appropriate sensor for feedback control and choosing a location for this sensor. Evaluating
different sensor options is based on various considerations such as the effect on actuator and stage dynamics, the effect on operating air gap, the relationship between
sensor output and actuator force, sensor noise, sensor bandwidth, and sensor reliability. In this section, we will address sensors that can provide estimates of the actuator
flux or actuator force. We do not cover current sensors here, since accurate current
sensing is assumed to be already available owing to its use with present state-ofthe-art Lorentz actuators. The types of possible sensors can be grouped according
the following classification scheme: 1) flux sensors; 2) force sensors; 3) acceleration
sensors; and 4) a combination of force sensors and acceleration sensors.
86
Flux Sensing
A sensor to measure magnetic flux density in the reluctance actuator air gap is required if the flux feedback topology shown in Figure 2-16C is chosen as the control
scheme. There are several advantages to flux sensing over the alternatives. The measured signal is bipolar in the case of a Hall-effect sensor or a sense coil, i.e., a negative
actuator flux results in a negative sensor output. This means that the sensor can distinguish between first- and fourth-quadrant operating regimes and the controller can
transition through the zero-flux point without being subjected to problems arising
from switching. That said, it is likely that the reluctance actuator will be operated
with a bias force and will be driven only in the positive flux region, so the sensor
bipolarity may be a moot point: because of the squared relationship of force with
flux, there is no need to operate in the negative flux region. The sensor location and
fixturing will likely have an insignificant effect on system dynamics, both in terms of
stiffness and added mass. Another advantage is that stage acceleration will not affect
the sensor measurement.
The main drawback of flux sensing is that the relationship between measured flux
and actuator force must be modeled accurately in order to achieve accurate force
control. This relationship, whether modeled analytically or implemented as a lookup
table, will likely depend on operating gap and possibly on rotation of the mover
relative to the stator.
Flux sensors can be categorized as either DC sensors or AC sensors.
(a) DC Flux Sensors: DC sensors include Hall effect sensors, magnetoresistive (MR)
sensors, and giant magnetoresistive (GMR) sensors. All have the capability of
measuring magnetic flux density at DC. This capability for direct flux measurement is its primary advantage over a sense coil, which is AC-coupled.
There are several disadvantages to direct-measurement sensors. One disadvantage is that the measurement is local: rather than measuring the average flux
density over the entire pole face area, the measurement is taken at only a small
fraction of the total pole face area. Since the flux density distribution over the
87
pole face will not be perfectly uniform, such a measurement will result in greater
sensitivity of the actuator force-to-measured-flux relationship to changes in the
flux distribution due to gap variation or off-axis relative motion between the
stator and mover.
The force-to-measured-flux relationship will also be more
sensitive to local variations in saturation of the magnetic material. One way
to reduce this effect is by using multiple sensors distributed across the surface
of the pole face, but this comes at the cost of higher reliability risk, additional
A/D channels, additional wires, and additional computation. Another difficulty
to direct-measurement sensors is that the sensor's placement is typically in a
high-risk location (on the pole face).
This also limits the minimum air gap
achievable and thus reduces the maximum force density capability of the actuator. One way to retain a small working gap is to recess the sensor in the pole
face, but this reduces the flux measured by the sensor relative to the flux in the
air gap.
(i) Hall-Effect Sensors: One possibility for sensor-based feedback control is to
incorporate a Hall-effect sensor into the actuator operating gap. Figure 218 illustrates a reluctance actuator with a Hall-effect sensor.
voltage,
VH,
The Hall
is proportional to both the current input to the sensor and
the magnetic flux density transverse to the sensor [91]. This voltage is then
the feedback variable used to compare against the desired flux density. One
downside of a Hall-effect sensor is that it is an active sensor: electronics
are necessary to provide a constant current to the Hall cell.
Hall-effect
sensors also typically have inferior noise characteristics and have higher
temperature sensitivity compared to a sense coil.
(ii) MR and GMR Sensors: MR sensors are based on the property of magnetoresistance that some materials exhibit. Magnetoresistance is the phenomenon whereby a material's electrical resistance changes in response to
an applied magnetic field. GMR sensors exhibit the same phenomenon,
but show a much larger percentage change in resistance for the same mag-
88
Figure 2-18: A reluctance actuator with a Hall-effect sensor in the air gap.
netic field. Like Hall-effect sensors, these sensors require a current or voltage source. The sensing resistors are usually configured as a Wheatstone
bridge. Downsides of MR and GMR sensors are that they typically have
a low linear range, they exhibit hysteresis, and they are usually unipolar,
i.e., one cannot distinguish between positive and negative magnetic fields.
(b) A C Flux Sensors (Sense Coil): A sense coil works by application of Faraday's
Law [39], shown here as
B - dS.
E-dl = -
(2.32)
This states that the line integral of the electric field around a closed contour is
equal to the negative rate of change of the surface integral of the magnetic flux
density passing through the contour. Equation (2.32) can be simplified to
c
-A
=
dttEA,
(2.33)
where A is the area of the surface and 13 is the mean flux density normal to the
89
surface.
Suppose that a coil with Ns turns is wound around the center pole face of the
reluctance actuator, as shown in Figure 2-19. Then, the area A that the flux
passes through is equal to NsAP, where AP is the center pole face area. The lefthand side of (2.33) is simply the negative of the electrical potential, or voltage
vs = NsAp
.
[39], between the coil terminals, denoted as vs. Equation (2.33) now becomes
Pdt
(2.34)
Figure 2-19: A reluctance actuator with a sense coil wound around the center pole
face.
Solving for P, we write
B =
A
vsdt.
(2.35)
The average flux density passing through the area enclosed by the sense coil is
proportional to the integral of the sense coil voltage.
A sense coil measurement offers several advantages over a direct-flux measurement. It is a more global measurement: since it is wound around the pole face,
the voltage at the terminals of the sense coil will be proportional to the aver-
90
age flux rate of change through the pole face. This results in lower sensitivity
of the force-to-measured-flux relationship to gap variations, to off-axis relative
motion between stator and target, and to local variations in saturation.
An-
other advantage is that a sense coil usually has better noise characteristics than
a Hall-effect sensor. A third advantage is that it is a simple passive sensor: it
only requires two leads to the coil output, whereas a Hall-effect sensor or MR
sensor requires additional electronics. Electronics may be needed to perform
the required integration, but this can be done remotely. Finally, a sense coil
does not take up real estate in the air gap, permitting the design of a more
efficient actuator and protecting the sensor from damage should a crash occur.
The main disadvantage of a sense coil is that it does not provide a true DC
measurement.
This results in drift when integrating the signal and poses a
significant challenge to obtaining an accurate estimate of the DC flux.
One
potential way to circumvent this problem is to estimate the DC flux from the
actuator current measurement. However, modeling the hysteresis between the
current and flux density is a challenge in its own right.
(c) Hybrid Configuration: A hybrid measurement scheme takes advantage of the
positive aspects of both a DC sensor and sense coil. One can use a Hall-effect
sensor for DC and low-frequency measurements and use a sense coil for highfrequency measurements. Katalenid in [49] demonstrates one method for implementing such a configuration.
There are three main disadvantages with this arrangement. First, extra sensors
result in extra wires and extra channels and more reliability concerns.
Sec-
ond, combining the two measurements into one flux estimate is not necessarily
straightforward since the two sensors will have different calibrations. The signal processing consequently becomes more complex.
Finally, we lose one of
the benefits of using a sense coil only: the air gap must be made large enough
to accommodate the DC sensor, thereby reducing the maximum force density
capability of the actuator.
91
Force Sensing
Force sensing is an alternative to flux sensing. It is required if the force feedback
configuration shown in Figure 2-16D is used. With force sensing, load cells (or strain
gauges mounted to flexures) measure a force proportional to their displacement. Force
sensing allows for direct sensing and control of the variable of interest, rather than
having to design a feedback loop around an intermediate variable such as flux. One
advantage this presents is that the need accurately to model the nonlinear relationship
between flux and force and gap is eliminated. Another advantage over flux sensing
is that one force sensor is capable of measuring the total actuator force, whereas one
flux sensor is only capable of measuring the flux in one air gap, despite the force being
dependent on the flux in each air gap. A final advantage to force sensing is that a
load cell will not be located in the air gap, unlike a Hall-effect sensor or MR sensor.
There are notable downsides to force sensing, however. Because of the fundamental F c B 2 relationship, the sensor is unipolar, i.e., the sensor cannot distinguish
the direction of flux. This can result in controller difficulties around the zero-flux
point. Another potential problem with a force sensor is that it adds another stiffness element in the force loop, which can lead to deteriorated system dynamics. A
third problem with force sensing is that load cells will not measure the true applied
force because the inertial force from the stage acceleration introduces an error into
the measurement. Compensating for this error is the most challenging difficulty with
force sensing. Finally, an actuator mechanical design that incorporates load cells and
is properly constrained presents challenges more complex than those associated with
flux sensing. As with flux sensors, force sensors can be categorized as DC force sensors
or AC force sensors.
(a) DC Force Sensors (Strain Gauge Load Cells): Strain gauges provide a DC force
measurement, and thus have a similar advantage over piezo-based load cells
(which are AC-coupled) to a Hall-effect sensor over a sense coil. Strain gauges,
however, have much lower stiffness than piezo-based force sensors, and this can
adversely affect the actuator or stage dynamics.
92
I
Another downside of strain
gauges is that they are sensitive to temperature. Strain gauge packages that
compensate for temperature are available, but they dissipate additional power.
A strain gauge measures deflection by measuring the change in electrical resistance of a material undergoing mechanical strain.
is usually measured using a Wheatstone bridge.
This resistance change
In a strain-gauge load cell,
the strain gauge is attached to some structure (such as a flexure) that deforms
linearly with load. The force can then be computed from Hooke's Law.
Figure 2-20 shows one potential actuator configuration that incorporates straingauge sensors mounted to flexures on the target side. The use of two strain
gauges permits the measurement of actuator force and one actuator moment.
One side of each flexure is attached to the target. The other side of each flexure
is shown connected to ground. In a real lithography machine, this connection
would be to the short-stroke stage instead.
SG2
SGJ
Figure 2-20: A reluctance actuator with strain guages mounted to flexures attached
to the target.
(b) AC Force Sensors (Piezoelectric Load Cells): Piezoelectric load cells offer the
advantage of having high stiffness and large dynamic range.
Piezoelectric materials accumulate electrical charge when stress is applied to
93
the material.
A charge amplifier measures this charge and converts it to a
voltage proportional to the applied load. Such a piezo-based force sensor cannot
measure true static loads because the charge accumulated from a static load will
gradually dissipate.
Figure 2-21 shows one potential actuator configuration that incorporates three
piezoelectric load cells mounted to the stator side.
In a scanning stage, the
other sides of the load cells would be connected to the long-stroke stage (not
shown in the figure). The three load cell outputs can be tied together so that
the voltage output of the charge amplifier is proportional to the sum of the
loads applied to each piezo cell. Alternatively, separate charge amplifiers can
measure the charge from each piezo cell independently, which would allow the
measurement of actuator force and two actuator moments.
Figure 2-21: A reluctance actuator with three piezoelectric load cells connected to
the stator.
Acceleration Sensing
Acceleration sensing is an alternative to force sensing. Instead of measuring the force
directly using force sensors, the force can be estimated from the stage acceleration.
The force on the stage is proportional to the stage acceleration. This includes components proportional to the snap and jerk (and higher-order derivatives) because of the
94
flexible-body dynamics of the stage. These components would have to be estimated,
possibly in a feedforward manner.
Accelerometers are small and can be mounted easily to the stage with very little
added mass. An added advantage is that an accelerometer will not affect the actuator
operating gap. Figure 2-22 shows an actuator with an accelerometer attached to the
target.
The most obvious liability of using an accelerometer for sensing is the one already
noted: it provides only an indirect measurement of the force. A second downside
is that the accelerometer must be attached to the short-stroke stage rather than to
the long-stroke stage to obtain a reasonable estimate of the force. The upshot is the
concomitant risks of disturbances from the additional wires and chuck distortion from
accelerometer power dissipation.
+VA
Figure 2-22: A reluctance actuator with an accelerometer attached to the target.
Force + Acceleration Sensing
Figure 2-23 shows an actuator with an accelerometer and three piezoelectric load cells
mounted to the stator. By using both force sensors and accelerometers, some of the
problems involved in using only one or the other method can be remedied.
95
An accelerometer permits compensation for the acceleration error due to force
sensing alone. Force sensing provides a direct measurement of the force, rather than
relying on force estimation required of acceleration sensing alone. Moreover, all the
sensors can be located on the stator, which is not feasible with acceleration sensing
alone.
The drawbacks of combined force and acceleration sensing include having to incorporate additional sensors and having to calibrate and combine different measurements
into an accurate force estimate. As with force sensing alone or acceleration sensing
alone, a combined force and acceleration measurement is a unipolar measurement.
Figure 2-23: A reluctance actuator with three piezoelectric load cells and one accelerometer mounted to the stator.
2.5.3
Advanced Control Techniques
Beyond standard feedforward and feedback control, other control techniques can be
used further to improve actuator performance. In this section, we will briefly describe
two of these: feedback linearization and iterative learning control (ILC). These techniques are used in conjunction with standard feedback and feedforward techniques
rather than in lieu of them.
96
Feedback Linearization
With feedback linearization, a nonlinear system is transformed into an equivalent
linear system by feeding back a measured variable as an input to the transformation.
This transformation permits the use of standard linear feedback control techniques.
For example, suppose y = P(v, x) is the nonlinear plant we are trying to control,
and we have access to a measurement of x. If P is invertible, then preceding the
plant with its inverse P-1 (y, x) will yield the linear system P-'P = 1. Figure 2-24
demonstrates the concept. The measurement x is fed back into the transformation
1
v = P-1 (u, x). Since v = P-1 (u, x) and y = P(v, x), then y = P (P- (u, x), x), and
the control signal u equals the output y. The controller C can be designed using
linear control techniques since the effective plant is now equal to 1.
10--
+
C
U
-*P-(u'x)
V1
-
P(v~x)
Linear
Nonlinear
Nonlinear
compensator
compensator
Plant
-
-
Figure 2-24: Block diagram of a controller with feedback linearization.
For some classes of nonlinear systems, feedback linearization is straightforward
and can easily be implemented in software or an FPGA. A possible implementation
of feedback linearization for a reluctance actuator controller is shown in Figure 2-25.
Here the actuator flux density is being controlled via feedback, using the topology
shown in the dotted blue box in Figure 2-16C. Figure 2-25 has an extra I(B, g) block
in the loop compared to the loop in Figure 2-16C. This is the nonlinear controller
that transforms the nonlinear system into a linear one. Note that because B depends
on the gap, a gap measurement input or gap estimate input is necessary for accurate
97
feedback linearization. If I(B, g) is a perfect inverse of B(I, g), the plant as seen by
the linear controller is equal to 1.
Linear
controller
Bd
Actuator
plant
Nonlinear
controller
+IB
I(Bg)
C
Actuator
current
B(Ig)
Actuator
flux density
Gap
measurement
Figure 2-25: Feedback linearization for a reluctance actuator.
For a reluctance actuator flux control, feedback linearization offers the potential for
effective control authority at both low flux levels and high flux levels near saturation.
Without feedback linearization, feedback control authority degrades at saturation
because the loop gain decreases.
The downsides of feedback linearization are the same as those for feedforward: it
requires an accurate plant model and is sensitive to changes in the plant.
Details
regarding feedback linearization as applied to electromagnetic actuators can be found
in [56, 85, 84].
Iterative learning control
Iterative learning control (ILC) is a technique that can be applied to systems with
repeating trajectories. Each time a trajectory is completed (called a trial or iteration),
ILC stores and uses information from that trial and previous trials to improve the
tracking performance for the next trial. ILC does this by effectively learning the input
required to drive the tracking error toward zero; as such, it changes the feedforward
signal from trial to trial. It can be thought of as a feedback controller in the iteration
domain: feedback after each trial is used to change the feedforward signal.
Figure 2-26 is a block diagram of the standard ILC topology [73].
The error
sequence during the kth iteration, ek, is passed through the learning controller L
98
and then added to the kth iteration feedforward input sequence,
the resulting combination through a 'robustness' filter
feedforward input sequence for the k + 1 iteration,
Q,
Uk.
After passing
the result is stored as the
Uk+1-
Q
L
Uk+1
Memory
Memory
ek
Uk
yy
es
-,
-1.-
C
L
Feedback
controller
Plant
Figure 2-26: Standard ILC configuration.
Assuming
Q
=
1 for the moment,
Uk+1 =
I
uk+1
can be written as
( LP
P)
1 +CP
Uk
+
L
1+T CP
r.
(2.36)
The optimal L is then given by the inverse of the process sensitivity function, L
(1 + CP)/P. Then
reference.
uk+1 =
P-r
=
and the output will perfectly track the desired
However, P/(1 + CP) is often not invertible over the entire frequency
band. In such cases, L is designed to approximate the inverse of the process sensitivity
function as well as possible. To ensure convergence of the sequence of inputs,
Uk,
the
criterion that must be satisfied is
I
L(s)P(s)
1 + C(s)P(s)
< 1
(2.37)
for any s = jw. To ensure that this requirement is satisfied at all frequencies, a
99
robustness filter
Q is added,
such that the stability criterion now becomes
Q(s)
for any s = jw.
(,._L(s)P(s)
I - 1 + C(s)P(s))
(s)P(s)
The robustness filter
frequencies below the cutoff. Designing
Q
< 1
(2.38)
is a low-pass filter with unity gain at
Q this way
ensures that at frequencies below
the cutoff, the performance of the learning controller will be unaffected.
ILC thus offers a straightforward way to improve the feedforward accuracy from
one iteration to the next. At frequencies below the crossover of the standard feedback loop, the gain of the optimal learning controller is dominated by the feedback
controller gain rather than the plant gain, so modeling the plant perfectly is less
critical than in the case of standard feedforward with no learning.
ILC can also
adapt to changes in the plant over time. However, it cannot adapt to non-repeating
disturbances. Further details on standard ILC can be found in [34, 73].
Other variations of ILC can be found in the literature. These include lifted ILC
[34, 73], current-iteration ILC versus past-iteration ILC [89, 17], and PD- and PIDtype ILC [63, 17]. Two good survey papers on ILC are [2, 17].
In this section, we have presented a brief overview of various control techniques
that can be applied to a reluctance actuator. In the next section we will shift our
focus from reluctance actuator control to variations on the basic reluctance actuator
configuration.
2.6
Alternate Actuator Configurations
Thus far in this chapter we have analyzed the reluctance actuator with a threepole stator. The most basic types of electromagnetic actuators include a single softmagnetic stator in the form of a 'U'-core (two poles) or an 'E'-core (three poles) with
one independent actuator coil. A number of variations upon this building block exist.
In this section we will consider a few of them.
100
2.6.1
Double-sided Actuator
A double-sided actuator, also described as a 'push-pull' pair, consists of a stator on
either side of the target. Such a configuration is shown in Figure 2-9. The primary
advantage with a push-pull pair is that the actuator module is bipolar: it can generate
force in both the positive and negative directions.
Figure 2-27 shows a chuck with 'pull-only' reluctance actuators and a chuck with
push-pull actuator pairs. The boxes marked 'x' and 'z' represent the x-Lorentz actuators and z-Lorentz actuators, respectively. With the pull-only configuration, the
entire y-force during acceleration is applied at only one end of the chuck. With
the push-pull configuration, the y-force can be distributed evenly among all four yactuators. This results in the y-force distribution being symmetric about the x-axis.
This leads to less distortion of the chuck and and is less likely to excite problematic
modes. An additional advantage is that the total target mass can be reduced since
each target in a push-pull configuration must only be capable of generating half the
force that a target in a pull-only configuration must generate.
Figure 2-27: LEFT: chuck configuration with one-sided (pull-only) actuators.
RIGHT: chuck configuration with double-sided (push-pull) actuators.
The major downside of a push-pull chuck configuration is that the chuck design
101
and long-stroke stage design become much more complex since the actuators interlock.
Designing the reluctance actuators to operate at small air gaps is also more challenging
because we now have to be concerned with tolerance stackups on both sides of the
target. Moreover, with a pull-only actuator, the target can be designed such that it
functions as the crash stop itself, for example by being preloaded with a soft spring
against a hard stop. In this way, during normal operation, the target has a rigid
connection because the spring pushes it against the hard stop and the stator generates
force on the target in the same direction as the spring. However, in the case
of a crash,
the target gets pushed against the soft spring, which will help absorb the energy from
the impact and protect the glass chuck from damage. Such a configuration is shown
in Figure 2-28. Alternatively, the hard stop and preload spring could be located on
the stator side. This concept has the potential to allow smaller operating gaps than
would be permitted otherwise since we do not have to allow additional air gap for
crash distance. In contrast, with a push-pull actuator pair, designing the target itself
to be the crash stop is less feasible since the target must be rigid in both directions
(although it may be possible to design the stators as crash stops). This means that
the air gap must be larger to avoid the stator hitting the target in the case of a crash.
Normal
Operation
Fr
r
TaF
Crash
Preload.
Spring
motion
T TarStage
Figure 2-28: LEFT: Target is preloaded with spring against hard stops for normal
operation. RIGHT: Spring allows target to function as soft stop in case of crash.
102
2.6.2
Multi-DoF Actuators
The actuator configurations shown thus far have only included one independent actuation coil per stator. This is the simplest configuration, but it only permits control
of the force in the drive axis. Owing to stage rotation, asymmetry, and other nonidealities in the actuator, the actuator in practice will exhibit stray torques about
both axes orthogonal to the drive axis. One way to mitigate these stray torques is
to make the actuator as compact as possible in these axes. Another way is to design
the actuator with multiple actuation coils and multiple sensors so that these stray
torques can be measured and controlled independently of the force.
First, we qualitatively demonstrate why a three-pole actuator is more susceptible
to parasitic torques than a two-pole actuator.
Consider Figure 2-29.
If there is a
rotation of the target relative to the stator, the effective air gap on one side becomes
smaller than the effective air gap on the other side. Since there are two separate
flux paths, more flux travels through the path with the smaller air gap, generating
more force on that side, and consequently, a torque. The relative sizes of the arrows
in the figure qualitatively demonstrate the amount of flux going through each flux
path. The air gaps in the two flux paths operate analogously to a parallel resistor
circuit where the resistances are changing: as one resistance gets smaller than the
other, more current will flow through that leg of the resistor network. Likewise, as
one air gap gets smaller in the magnetic circuit of the stator core (and thus, the
corresponding reluctance gets smaller), more flux will flow through that leg of the
magnetic circuit.
A two-pole actuator does not have this problem since there is only one primary
flux path. If there is a rotation of the target relative to the stator, all the flux must
still go through the same path and so to a first-order approximation there are no
parasitic torques produced.
If the three-pole actuator includes two independent actuator coils, this torque can
be controlled. Figure 2-30 shows what this configuration might look like. The two
independent flux paths are driven with amp-turns NI1 and NI 2 . The flux paths
103
Figure 2-29: A three-pole actuator showing the effect on flux distribution of a relative
rotation between the stator and target.
reinforce in the center leg.
The flux in the left and right air gaps can be driven
differentially to generate a torque to counteract any parasitic torque that arises. For
this 2-DoF actuator to work effectively, both the force and torque must be sensed
or estimated, thus requiring two independent sensors. This could take the form of
two Hall-effect sensors or sense coils, one on the right pole and the other on the left
pole, for example, or the stator could be mounted on two or more load cells. Another
possibility would be to measure both the left and right air gaps and estimate the
force and torque from the gap measurements and actuator currents. These sensor
configurations are not shown in the figure.
I I
Flux path 2
Flux path 1
Figure 2-30: A three-pole actuator with two independent actuator coils.
104
Other multi-DoF actuator configurations are also possible. For example, the configuration presented above could be adapted also to control the torque about the
x-axis by adding a second set of poles and actuator coils in the out-of-plane direction.
2.6.3
Multi-pole Actuators
An alternative to the two-pole or three-pole stator designS is a stator having additional poles. Additional poles allow for a smaller target mass and stator mass for the
same force capability [561. The reason for this is that by splitting the flux among additional poles, the minimum thickness of the target needed to prevent flux saturation
can be made smaller. Likewise, the minimum thickness of the stator back leg can be
made smaller.
For example, consider the case of a three-pole actuator versus a two-pole actuator.
In the three-pole actuator, there are two primary flux paths as shown in Figure 2-30.
The flux from the center pole face is split between the two sides. In contrast, in a twopole actuator, all the flux travels in one path through the target. The upshot is that
the target in the three-pole actuator will not saturate as quickly and can therefore be
made thinner. Since force is proportional to pole face area, the three-pole actuator
will have the same force capability as the two-pole actuator if the total pole face
areas of each are identical. Thus, the target for a three-pole actuator with identical
force-generating capability to a two-pole actuator can be made significantly lighter
than the two-pole actuator target. This phenomenon can be adapted to multi-pole
stator designs to decrease the target mass further if desired.
Figure 2-31 shows a
five-pole actuator.
Drawbacks of multi-pole actuators include increased complexity and reduced compactness in the horizontal direction, which can increase stray torques. However, the
additional coil(s) in a multi-pole stator design permit the independent control of both
force and stray torque.
105
Figure 2-31: A five-pole actuator with two independent actuator coils.
2.7
Summary
In this chapter, we developed the fundamental electromagnetic theory underlying the
reluctance actuator. By comparing the reluctance actuator operation with the Lorentz
actuator operation, we learned that the reluctance actuator can generate higher forces
at small air gaps, but that it suffers from higher stiffness. We gave an overview of core
materials and learned that CoFe is likely the best candidate for a high-force-density
actuator owing to its high saturation flux density. We presented an overview of the
nonlinearities inherent to a reluctance actuator and reviewed techniques with which
to linearize the actuator through actuator configuration and control. We learned that
an air gap tends to linearize the actuator by 'shearing' the hysteresis curve. We will
make use of this fact in Chapter 6. We determined that flux control is an attractive
controller topology, as it eliminates a number of nonlinearities present with current
control but does not suffer from the problems intrinsic to force sensing. Finally, we
presented some variations on the basic reluctance actuator configuration.
In the next chapter, we focus on the reluctance actuator hysteresis. We present
methods for modeling ferromagnetic hysteresis. These models will ultimately be used
for real-time flux estimation and flux control to linearize the actuator.
106
Chapter 3
Hysteresis Modeling
In this chapter, we present approaches for modeling ferromagnetic hysteresis.
The
purpose of these models is to incorporate one of them into a full reluctance actuator
model that can then be used both to predict and to investigate actuator behavior and
to provide an accurate estimate of the air gap flux density for real-time control. In
this chapter, we examine only the basic hysteresis model itself. Incorporation into a
full actuator model will be investigated in subsequent chapters.
First, we present a general overview of hysteresis phenomena and review physical
explanations for hysteresis in ferromagnetic materials. We then review the various
types of hysteresis models in the literature.
Next, we examine the Classical Scalar Preisach hysteresis model in more detail.
We first define the model in mathematical and in graphical terms. We then provide
two alternative methods for identifying and implementing the Preisach model. We
present simulation results showing hysteresis loops generated by the Preisach model.
Finally, we investigate an alternative model to the Preisach model: the Chua
model. We show how to identify the Chua model and present simulation results of
hysteresis loops generated by the Chua model. We identify drawbacks and benefits
of the Chua model in comparison to the Preisach model.
107
3.1
Introduction to Hysteresis
Hysteresis is a nonlinear phenomenon in which the output of a system depends not
only the present input, but also on the prior history of the input. This results in
a lag between the input and output; unlike with dynamic systems, this lag persists
even at quasi-static conditions. Hysteresis manifests itself in a wide variety of phenomena: these include ferromagnetism, piezoelectricity, friction, magnetoresistance,
mechanical deformation, and many others.
Ferromagnetic hysteresis can be described at the molecular level [14] in the following way: when an external magnetic field is applied to a ferromagnetic material,
the atomic magnetic dipoles will attempt to align themselves with this field; however,
when the field is removed or reduced, some of the dipoles will retain their alignment.
This results in the magnetization curve following a different path as the external field
is reduced [14]. Figure 3-1 illustrates how this curve might look. As the applied magnetic field is increased from its initial state a, the magnetization follows path 1. As
the applied magnetic field is decreased from state b, the magnetization follows path
2.
At the macroscale, the physics of ferromagnetic hysteresis is more complex [14].
A typical ferromagnet is comprised of a number of magnetic domains that interact.
Within each of these domains, the magnetic dipoles are aligned and the domain
has a uniform magnetization; however, the magnetization of each domain will vary
in magnitude and direction.
When the external field is varied, the domain walls
separating adjacent domains move, causing the domains to change shape. Domains
with a magnetization direction close to the direction of the external field expand while
domains not aligned with the external field shrink and will ultimately disappear if the
external field reaches sufficiently high values [14]. When the external field is reversed,
domains with reversed magnetization will form. These will get larger and larger as
the field continues to reverse [14].
108
M
2
b
a
H
Figure 3-1: Hysteresis phenomenon, shown as magnetization versus applied field.
3.2
Prior Art
Much research has gone into modeling hysteresis. Hysteresis models can be classified
as either physical models or phenomelogical models. Physical models are based on the
underlying physical principles that give rise to the hysteresis. The most widely known
example of a physics-based model applied to ferromagnetism is the Jiles-Atherton
model [47]. Physics-based models applied to piezoelectric behavior are documented
in [1, 51, 76]. One downside of physical hysteresis models is that they are specific to
the type of hysteresis being modeled.
Phenomelogical models are models that attempt to model the hysteresis mathematically without regard to the underlying physical principles that give rise to the
hysteresis. These types of models can be sub-classified into integral models and statespace models [49]. Integral models take a number of weighted fundamental operators
the number of
(such as a relay) summed in parallel that each act on the input. As
these operators approaches infinity, the summation becomes an integration. The most
widely used integral-type model is the Preisach model [61]. Other integral-type mod-
109
els include the Prandtl-Ishilinskii model [55] and the Generalized Maxwell-slip model
[3]. State-space models use a set of differential equations with nonlinear functions to
model the hysteresis. State-space models include the Chua model [25, 26, 23] and
the Duhem model [88]. Other state-space models that are special cases of the Duhem
model are the Bouc-Wen model [45, 77], the Dahl model [31], the LuGre model [32],
and the Coleman-Hodgdon model [27, 87].
Phenemological models tend to be more generalizable than physical models. Among
phenemological models, state-space models are simpler and more computationally efficient. Integral models are more computationally intensive, but have the advantage
of greater capability for high accuracy.
3.3
The Classical Scalar Preisach Hysteresis Model
The Preisach hysteresis model is well known and is the model most commonly used
to model ferromagnetic hysteresis.
It is capable of high numerical accuracy and
prediction [9]. Various modifications can be made to the basic Preisach model for
improved accuracy. Some of these modified models have gained sufficient prominence
to earn their own names: these include the Modified Preisach Model [20], the Moving
Preisach Model [33], and the Product Preisach Model [33]. The basic scalar version
of the Preisach model can also be generalized to vector models [61, 33]. The basic
building block for the Preisach model is a non-ideal relay, called a 'hysteron' in the
literature. Figure 3-2 is a graphical representation of one such hysteron. The input
of the hysteron is u and the output is v. The output can take one of two values: 1 or
-1.
Switching occurs at the threshold values of u (a or
) depending on the state of
the input.
This can be described mathematically as
V(U) =
1,
if U > a
--1,
if
<
k,
if
< U < a,
110
(3.1)
V
u
fa
Figure 3-2: A Preisach hysteron with the threshold values a and 0 for switching
output states.
where k = -1
if
UT
< a, and k
1 otherwise, where
=
UT
is defined as the value of u
the last time that u was outside the thresholds (i.e., u > a or u < 0).
Suppose we now have multiple hysterons connected in parallel, each one denoted
by Resoi.
Re,,
has threshold values ai and Oi. Each hysteron is multiplied by a
weight pi and the outputs are summed. Figure 3-3 illustrates this structure.
The
result is a discrete form of the Preisach model, and can be expressed as
v
=
(3.2)
pi Rai,3 , (u (t)).
As the number of hysterons approaches infinity, the Preisach model becomes continuous, and can be written as
v(t) =
f
pjpR (u (t)) dad.
(3.3)
The integration in (3.3) takes place over the space of a, / values. This way we account
for the state of every hysteron, which is defined by its threshold values of a and /.
111
Ra#
UJ
2
2
:R a
"9
Ra
Figure 3-3: A parallel connection of weighted hysterons.
The integration is restricted to cases where a > 3 because R, 3 is not defined for
,3 > a. This model is known as the Classical Scalar Preisach Model (CSPM). The
weighting function pcu3
p(a, 3) is known as the Preisach density function (pdf).
Setting up the Preisach model involves identifying and specifying this function. It
can be specified with an analytic function (continuous model) or numerically with a
lookup table (discrete model) and can be found experimentally.
While the Preisach model is a phenomenological model, the summing of hysterons
in parallel with different switching levels bears resemblance to the macroscopic behavior of magnetic domains within a ferromagnetic material [61].
3.3.1
Graphical Interpretation of the CSPM
Details of a graphical perspective of the CSPM can be found in [61]. The CSPM can
be interpreted graphically in the following way. Consider Figure 3-4, which shows the
a-0 plane, also known as the Preisach plane. The area over which the integration in
3.3 occurs is marked by the red triangle, defined by the lines a
112
-
3, a = u, and
#
= -u,.
The variable u, is the maximum value of the input. In a Preisach model
of magnetic hysteresis, this would correspond to the largest magnetic field intensity
expected, typically the magnetic field intensity H, at which saturation occurs. The
red triangle represents the pdf p,3. Each point (ai,#3i) within the red triangle has a
value pi. The pdf is assumed to have a value of zero for points (a, 0) that lie outside
the red triangle.
a=f3
a
uu,
S' S
Figure 3-4: The Preisach plane.
Suppose now that the initial state of the CSPM is -u,, -v,, where v, is the
negative saturation value of the CSPM output. This state correpsonds to the lowerleft vertex of the red triangle. As u(t) is increased from u, to some value u1 , all
the hysteron operators Raf with thresholds a less than ui are switched to
+1. This
subdivides the triangle into two areas, denoted by S+ and S- in Figure 3-5A. S+
corresponds to the (a, 3) points where the Raf3 operaters have a +1 output, and Scorresponds to the (a, /) points where the ROf operators have a -1 output.
This
.
subdivision corresponds to the line a = u1
Now suppose that we decrease the value of u from u1 to U2 . All R.0 operators
with thresholds
/
above u 2 will now switch to output -1. Note from Figure 3-5B that
the subdivision between S+ and S- now has a vertical link corresponding to the line
113
u2
.
3=
Next suppose we increase the value of u again from u 2 to U3 , where u3 < u. Those
Ra operators with thresholds a less than U3 will switch back to output +1. A new
horizontal link in the subdivision between S+ and S- is formed corresponding to the
line a
=
u 3. See Figure 3-5C.
Finally, suppose that we increase u further to a value u 4 , where u 4 > u1. This is
shown in Figure 3-5D. All Rf-operators with thresholds a less than u 4 will switch
Note that subdivision between S+ and S- is once again completely
horizontal and now coincides with the line a
=
u4
.
to output +1.
What the foregoing analysis demonstrates is that the Preisach plane is always
subdivided into two regions: S+ and S-. In the S+ region, all Rap operators are
switched to the +1 value; in the S- region, all R,, operators are switched to the -1
value. The line subdividing the two regions consists of vertices that correspond to the
previous local minima and maxima of the input. When the input is increasing, the
final link connecting the subdividing line to the a =
/
line is horizontal and coincides
with the line a = u(t). When the input is decreasing, the final link connecting the
subdividing line to the a =
/
line is vertical and coincides with the line
/
=
u(t).
The integral in (3.3) can be expressed in terms of S+ and S- as
V(t) =
f
p +)yR (u (t)) dado +
ff
[LMpRafl
(u (t)) dad3. (3.4)
Since RO (u (t)) = 1 in the region S+ and Rp (u (t))
-1 on the region S-, we
can write (3.4) as
v(t) =
Jp+
MPodad -
pfS
/IMPodad3.
(3.5)
It is clear then that v(t) depends on the particular subdividing line between S+ and
S-. In particular, it will depend on the past local minima and maxima of the input
that define the vertices of the subdividing line. In this way, we see graphically how
the Preisach model output depends on the input's history. The output v can thus
114
B.
A.
a
a
I
__ _ _ _ _ _ _ _
U2
Ai
/Z
1
1
Li
- U
S+
s+
D.
C.
a
a
8-
8U2
U1
flS+U3
SU1
3+
/
-
U3
I
.
.
Figure 3-5: A: Input u(t) increased -u, to u 1 . B: Input u(t) decreased from ui to u2
C: Input u(t) increased from u 2 to u3 . D: Input u(t) increased from U3 to u 4
have different values for the same input value u because the subdividing line is not
unique for a given u. Graphically, the non-unique subdividing line is the result of the
two different rules that govern the subdividing line's progression: when the input is
increasing, the subdividing line is characterized by a final link that is horizontal and
moving upwards. When the input is decreasing, the subdividing line is characterized
by a final link that is vertical and moving leftward.
Finally, note from Figure 3-5D that once the input increased beyond u1 , the verof
tex in the subdividing line at (u2 , uI) was wiped out. This is a general property
the Preisach model, and is aptly named the 'wiping-out' property. The wiping-out
an
property results in the removal of every vertex on the subdividing line that has
a-coordinate below the current input or a -coordinate above the current input. One
115
implication of this fact is that in a subdividing line with multiple vertices, any particular vertex will always have an a-coordinate less than that of the vertex immediately
to the left of it and a
-coordinate greater than that of the vertex immediately above
it. Graphically, this means that the 'staircase' that the vertices form on the Preisach
plane will always descend from left to right and never from right to left. See Figure 36 for an illustration. A further implication is that when implementing the Preisach
model, only those local minima and maxima that have not been wiped out must be
stored in memory.
S_ Subdi tiding line
S
Sduiv id in gk lin e
S+
3F-Z
S+
Possible
Impossible
Figure 3-6: LEFT: A possible instantiation of the Preisach plane subdividing line.
RIGHT: An impossible instantiation of the Preisach plane subdividing line.
3.3.2
Identification and Implementation of the CSPM
Now that the CSPM theory has been developed and explained, the next step is to
find methods for identifying the model and implementing it. What this boils down to
is identifying u,,3. There are various methods for doing this, and in this section, we
will briefly review some of these methods in the literature before giving more detailed
presentations of two of them.
In [61], pt
is identified by using the first-order reversal curve method. A first-
order reversal curve is obtained by increasing the input u from negative saturation u,
to some value a before then reversing the input and decreasing u to some value
116
f.
The
values a and 0 define the first-order reversal curve, so-called because it corresponds
to the first reversal of the input. This process is repeated for different values of a and
/
until we have a series of first-order reversal curves ranging between -u, < a < u,
and -- u, <
/
< a. The output values v are recorded for each first-order reversal
curve, each corresponding to a different state on the Preisach plane. We can then
apply an algorithm to populate the Preisach plane with ucfl values at discrete points
using the data obtained from the first-order reversal curves. Interpolation is used to
determine p
values that lie between the discrete points. One downside of the first-
order reversal curve method is that the most straightforward algorithm to populate
the Preisach plane requires taking second differences of the measured data. However,
in [61] an alternative algorithm is developed that avoids this. In [12, 11], A"3 is
identified by using the centered cycles method. Here, a series of B-H loops each with
different amplitudes is experimentally recorded where H and B are odd-symmetric
about the origin. This can be done for instance by driving H with a sine wave centered
around zero. From this data, we can populate the Preisach plane. Two benefits of
this method over the reversal curve method are: 1) the tests required (symmetric
B-H loops) are easier to obtain experimentally; 2) the algorithm used to populate
the Preisach plane is less sensitive to measurement noise because it does not require
computing second differences of the data. Cardelli in [20] uses a similar method of
symmetric minor loops to identify pAt . In [7], pa is assumed to be identical to the
product of two single-variable functions. This simplifies the analysis, and allows the
use of only the virgin curve and the major loop for identification of Au'. The virgin
curve is the path followed by the magnetization when it starts from the demagnetized
state [16]. The major loop is the M-H or B-H loop traversed when M is cycled
between negative saturation and positive saturation. The virgin curve and the major
loop are indicated in Figure 3-7.
Della Torre Implementation
Rather than explore the entire Preisach plane experimentally, which can be tedious,
Della Torre noted that in many magnetic materials, pt3 can be approximated with
117
M
Virgin curve
H
Major loop
Figure 3-7: Virgin curve and major loop of the hysteresis curve.
a simple function. In hard magnetic materials, this function often takes a Gaussian
form [33].
In soft magnetic materials, this function is better approximated with a
Lorentzian function [9].
In soft materials, magnetization is mainly determined by
domain wall motion, and this motion is typically unidirectional with a monotonically
increasing or decreasing applied magnetic field. This means that we can model the
probability distributions of a and 3 as statistically independent. This permits us to
write p,, as the product of two identical single variable functions [9], or
y (a, #)
=
y, (a) /pt, (-0)
(3.6)
Switching the input variable from u to H in order to represent applied magnetic field,
we can write pu(H) as a Lorentzian function,
ps(H)
Ad
1+
2
(3.7)
HcH
where Ad, H, and a- are the parameters to be identified. One advantage of the Della
Torre implementation is that it only requires identifying five parameters (in addition
to the three parameters above, the magnetic field at which saturation occurs, H, and
saturation magnetization, M, must be identified). If we replace the output variable v
with M to represent magnetization, we can differentiate (3.3) with respect to H and,
118
making use of (3.6), write
2pd(H)
JH
uS(--0) d3,
(H) =
dH
-
if
> 0
(3.8)
0
2,(-H)
u(a)da,
JH
if-
dt
<0.
Here, HO is the last reversal point. The first equation applies to the case when the
input H is increasing, while the second equation applies to the case when the input
H is decreasing. The factor of 2 arises from the fact that the change in magnetization
when a hysteron switches from its negative value to its positive value is twice its
magnitude.
Substituting (3.7) into (3.8) for the case of an increasing field, we write
2Ad
dH
I
Ad
jH
( H-He
Ho
d07
1
+
dM (H)
H.
H,
Aae-p-H
[HHJ]
[
~arctan(
+ (H-Hg~)
S HH
1 +cH
1+
dM
dH
(H)
2A2d
2
H-H
2
+ (H-He
HC
-
H + He
HC
arctan
2
Ho
artn(H
H)
-HHc
- H c +arctan
HC- -- arctan
2A 2
=
H
-Hc
-arctan
-- HoH - H
Ho + He
a-)
He
(
2Ad
(3.9)
Similarly, for a decreasing field, we derive
dM (H)
dH
2A 2
=
+
HH
-
He arctan
H
Ho - He
(-
0-
HU)2H
arctan
H - He
Hc
(3.10)
Here we make a quick aside to discuss a subtlety with the Preisach model that
occurs when rather than traversing the Preisach plane from a reversal point, we
traverse it from the demagnetized state. In this case, assuming that the hysteresis
function is symmetric, the subdividing line between S+ and S- is no longer a staircase119
like structure comprised of vertical and horizontal links, but is rather a line coinciding
with the line a = --.
is symmetric, p(a,
3)
The reason for this is that when the hysteresis function
=
p(-O, -a),
so that the pdf above the subdividing line is
identical the pdf below the subdividing line. From (3.5), the output v(t) will thus be
equal to zero. Because of the symmetry of p3 about the a = -0 line, the output
for a decreasing input from the demagnetized state will be equal and opposite to the
output for an increasing input from the demagnetized state. This matches expected
behavior for traversal from the demagnetized state for a symmetric hysteresis function.
Figure 3-8 shows the Preisach plane in the demagnetized state.
a
Figure 3-8: The Preisach plane showing the demagnetized state.
Increasing the input from the demagnetized state now results in a subdividing
line that looks like the one shown in the left illustration of Figure 3-9. Decreasing
the input from the demagnetized state results in a subdividing line that looks like
the one shown in the right illustration of Figure 3-9. As before, the last links of the
subdividing line in the increasing and decreasing cases are horizontal and vertical,
respectively; however, the rest of the subdividing line remains coincident with the
a = -3
line.
120
L>
H
a
'4
'4
'4
'4
'4
Figure 3-9: LEFT: Preisach plane with H increased from the demagnetized state.
RIGHT: Preisach plane with H decreased from the demagnetized state.
The different structure of the subdividing line changes the integration limits in
3.8. Instead of the last reversal point HO as one of the limits, the corresponding limit
This can be understood graphically by reference to Figure 3-9.
is replaced with -H.
In the left-side figure when H is increasing, the new area (marked by the dashed green
triangle) added to S+ is limited along the 3-axis by the lines a = -3
and a =
/.
Since a equals the current input H, the lower and upper limits when integrating
along 3 are therefore -H and H, respectively. Likewise, in the right-side figure when
H is decreasing, the new area (marked by the dashed green triangle) added to Sis limited along the a-axis by the lines a
=
This corresponds to
3 and a = -3.
lower and upper limits when integrating along a of H and -H, respectively. When
beginning from the demagnetized state, (3.8) thus becomes
2pd(H)
H
uS(-0) dO,
dH
-H
2p(-H)]
1
>0
if d
(3.11)
dM(H) =dH
us(a) da,
if
-H
<0.
H
Equation (3.9) becomes
dM
dH
(H)
1-
2A2
__-d
+
H-He 02
H(
-
arctan
Hi+He
- - arctan
H
H
,
-H + H
7)o
H2
(3.12)
121
and (3.10) becomes
dM
dH
=
2A2
1+
HH,
He
2)
-
arctan
r(H)
-H - H
He
H
-
arctan
H - Hc
He
(3.
(3.13)
To implement the Della Torre model, we use (3.9)-(3.10) and (3.12)-(3.13).
One
way to implement the Della Torre model for M as a function of H is to integrate
these equations analytically with respect to H. This integral can be solved in closedform [9]. However, the equations that result are very lengthy and unwieldy. More
importantly, the evaluation is no longer accurate once the last extrema Ho is wiped
out.
Another way to implement the Della Torre model-and the one pursued here-is
to evaluate dM/dH rather than M directly and then to integrate numerically. The
advantage of implementing the Preisach model in this way is that this evaluation
involves only one computation, whereas implementing the Preisach model in double
integral form involves on the order of N 2 /2 computations, where N is the number of
vertices of local extrema in the subdividing line [86].
We implemented the Della Torre model using a MATLAB function we wrote.
Values for He, a, Ad, H,, and M, are specified within the function. The function takes
as inputs the current value of H, the previous value of H, a variable indicating whether
the input is increasing or decreasing, a stack that stores the previous local minima
and maxima, and a variable indicating whether or not the initial state of M is the
demagnetized state. We create a vector of the input signal H and then loop through
this vector, computing dM/dH for each value of H using the MATLAB function.
The value of M is then computed by integrating dM/dH numerically after each step
and adding the result to the previous computed value of M.
Besides the current
value of dM/dH, the function also outputs an updated stack of the previous minima
and maxima and the updated variable indicating whether the input is increasing or
decreasing.
When the inputs are passed into the MATLAB function, the function first de-
122
termines whether the input is increasing or decreasing and whether or not we are
traversing from the demagnetized state.
Based on this evaluation, it determines
which equation among (3.9)-(3.10) and (3.12)-(3.13) to use. The function then updates the stack of previous minima or maxima. If the current input is increasing and
larger than the previous maximum or decreasing and less than the previous minimum,
the function removes the last two extrema from the stack. The function continues
to go through the stack to remove any other extrema that should be 'wiped-out'. If
the input has just reversed direction, the function adds the last reversal point to the
stack. The function then sets HO equal to the last reversal point of the updated stack
if (3.9) or (3.10) are being used to evaluate dM/dH.
It evaluates the appropriate
equation and then outputs dM/dH.
Identifying the Parameters of the Della Torre Model
Identifying Ad, He, and a is straightforward. The variable H, is related to the width
of the hysteresis loop and for many materials is very close in value to the coercivity of
the material Hc. If we inspect (3.9) for the major ascending branch of the hysteresis
loop (where HO is replaced with -H, the negative saturation input), we can reason
that for values of a > 0.5, the maximum value of dM/dH occurs close to H = He. It
is easy to see that maximum value of the term outside the brackets occurs at H = Hc.
The bracketed term will asymptote toward the constant maximum value of
7r
as the
value of o- is increased. Therefore, for values of a large enough, the maximum dM/dH
will occur near H = H,. We can thus often approximate H, either from the coercivity
of the material, or, if we have the M-H data for the major hysteresis loop, with [9]
dM a
dH )H-H
dM a
(3.14)
where the superscript a denotes the ascending branch of the major hysteresis loop.
In other words, if we compute dM/dH from the M-H data, then we can identify H,
by locating the value of H where the maximum dM/dH occurs. Figure 3-10 shows
three hysteresis loops constructed from the Della Torre implementation, each with
123
different values of Hc. For each loop, oM = 0 occurs at H
20 and pOM, = 1.5 T. It can be seen that
He.
2
0
-1-H
-
-2
-20
-10
0
H [A/m]
10
2 A/m
HC = 5 Am
H =10A/m
C
20
Figure 3-10: Three hysteresis loops constructed from the Della Torre CSPM with
different values of H.
The parameter a is related to the slope of the major hysteresis loop at H = Hc.
From [9], o- can be calculated from
a~
(dM)
d
Md
dH
J H=H,
(H+HO)
arctan (2a) - arctan (H+
arctan
(2)2
(3.15)
H-HeL
(
He
where the superscript d refers the descending branch of the major hysteresis loop.
We can solve for a for instance by plotting the right-hand side as a function of o- and
locating the value of a where the graph equals the value of left-hand side. The points
on the major loop needed to evaluate the left-hand side are shown in Figure 3-11.
Because (dM/dH)_H=
>> (dM/dH)dH, the left-hand side will be much greater
than 1.
If a >> 0.5, we can approximate (3.15) as
r ~ 2 [1 + (2o-) 2 ]
,
(3.16)
where r is the value of the left-hand side from (3.15). We can solve this equation
124
M
dH
evaluated here
d
H=,
H
evaluated here
(1dHH=H,
j
Figure 3-11: The dM/dH values needed to evaluate a.
explicitly for a as
o2
1
r
(3.17)
-1.
Figure 3-12 shows three hysteresis loops constructed from the Della Torre implementation of the CSPM, each with different values of a. For each loop, H, = 5 A/m and
MoM, = 1.5 T.
1. 5
0. 5
-
0
-0.51
=
-___
5
-20
=2
=20
-10
0
H [A/m
10
20
Figure 3-12: Three hysteresis loops constructed from the Della Torre CSPM with
different values of a.
125
Finally, the value of Ad can be calculated using results from [9] via
MS
Ad =
HfH
1+(a-c
f
a(
-H
(3.18)
d
1+( Oc
)da
Figure 3-13 shows the minor loops that the Della Torre implementation can produce.
These plots were simulated using the MATLAB function described earlier. We used
input H vectors with smaller amplitudes to simulate the minor loops.
I
1.5
1
0.5,
0-
74
-
-0.5
-
1 -0 L.-20
'
-
-1
-10
0
H [A/m]
10
20
Figure 3-13: Minor loops constructed from the Della Torre implementation of the
CSPM with parameters H, = 5 A/m, o = 20, poM, = 1.5 T.
Hui Implementation
In [44], a method for identifying and implementing the CSPM based on data obtained
from the major hysteresis loop is presented. Here, the output is flux density, B, rather
than M, so this terminology is followed in the subsequent discussion.
First, the function T(H1, H2 ) is defined as
T(H1 , H2)
=
1111"
(a, )dad.
(3.19)
In the Preisach plane, this corresponds to integrating over the right triangle with
vertex corresponding to a = H1 and 3 = H2. See Figure 3-14.
126
H
T(HI ,J
H2
Figure 3-14: The area of the Preisach plane corresponding to T(Hi, H2 ).
Suppose now that we have reached positive saturation (H,, B,) and now decrease
the magnetic field to some value H. This corresponds to traveling along the descending branch on the major hysteresis loop. The Preisach plane corresponds to that
shown in the left illustration of Figure 3-15.
aa
H,
H
Ss+
Figure 3-15: LEFT: Preisach plane corresponding to the descending branch of major
hysteresis loop. RIGHT: Preisach plane corresponding the ascending branch of the
major hysteresis loop.
127
The flux density corresponding to this state can be written as
Bd (H) = fIOpdad - fpfdaod,3,
=
(B.-T(H,,H))-T(H,H),
Bd (H) =
B,-2T(H,H).
(3.20)
Likewise if we start from negative saturation and increase the magnetic field to some
value H, the Preisach plane configuration depicted in the right illustration of Figure 315 obtains. We can write Ba(H) as
B"(H) = -B, + 2T(H, -H,).
(3.21)
We can generalize this analysis to determine the flux density for a magnetization
curve that has n reversal points:
Bn+2TH,Hn), if
B(H) =
Bn - 2T(H, H),
if
dH
dH
-
dt
>0
;H
(3.22)
< 0.
The variables Hn and Bn are the magnetic field and magnetic flux density, respectively, of the last reversal point. From (3.22), we see that the parameter we need to
identify in order to implement the Preisach model is T(H1 , H2 ).
In [44], it is shown
that T(H1 , H2 ) can be written as
T(Hi, H2) =
+ F(H)F(-H2),
2
(3.23)
where
H,)
F(a) =
a
pst(a)da.
(3.24)
Here we have again assumed that p,3 is separable into two identical single variable
128
functions as was done for the Della Torre implementation.
The function F(a) can be expressed in terms of the ascending and descending
branches of the major hysteresis loop as
Bd(a) - Ba(a)
F(a) -
2
Bd(a)
Bd(-a),
if a > 0
'~(3.25)
if a < 0.
Finally, in [44] the initial flux density curve is derived in terms of F(H) and F(-H)
as
Bz(H) = [F(-H) - F(H)] 2 ,
(3.26)
where the superscript i denotes the initial curve.
We can see then that (3.22) and (3.26) are ultimately reducible to terms that
involve only the ascending and descending branches of the major hysteresis loop. To
identify the CSPM using this method then, we need only experimentally to determine
the major hysteresis loop.
This data can then be implemented as a lookup table
for real-time operation, or an analytic function can be fitted to the ascending and
descending branches of the major loop.
The algorithm for implementing this form of the CSPM is simple. We create a
stack for storing the previous minima and maxima. We update the stack by checking
the stack against the current input and removing any previous extrema that should
be wiped out. If the input has just reversed, we add the new reversal point to the
stack.
After updating the stack, we check the last reversal point.
If the stack is
empty, we use (3.26) to calculate B. Otherwise, we use (3.22) with the last reversal
point for H,. Figure 3-16 shows the major loop as well as minor loops constructed
with the Hui implementation.
Accounting for Assymetry in the Hui Implementation
The foregoing analysis assumed that the hysteretic behavior is symmetric with respect
to the origin of the B-H plane. In reality, this will not always be the case.
129
1.5
1
0.5
0
-0.5
-1
-1.5'
-20
-10
0
H [A/m]
10
20
Figure 3-16: Major loop and minor loops constructed from the Hui implementation
of the CSPM.
The equations used to implement the CSPM can be modified to account for asymmetry. We replace the function F(a) with two functions F+(a) and F-(a). F1 (a) is
identical to F(a) given in 3.25. F-(a) can be written as
-Ba(a) + Bd(a)
F-(a) =
if a > 0
22- Ba(-a)
-Ba(a),
(3.27)
if a < 0.
T(HI, H2 ) is modified as to
Ba(H1 )
T(H1, H2 ) =
Bd(H 2 ) + F+(H )F+(-H
1
2 ),
2
Ba(H2 ) - Bd(H1 ) + F-H1 )F(-H2 ),
2
-
if H > 0
(3.28)
if H < 0.
Finally, the equations for calculating B(H) are modified as follows. For increasing
H, we write
B(H)
if H > 0
Bn + 2T(H, H),
Bn + 2T(Hn, H), if H < 0.
130
(3.29)
For decreasing H, we write
Bn-2T(H, H),
B, - 2T(H, H,),
B(H)
(3.30)
if H>O
if H < 0.
And finally, for the virgin curve, we write
[F+( _-H) - F+(H)]2 ,
-
3.4
[F-(-H) - F-(H)] 2 ,
if H > 0
if H < 0.
The Chua Hysteresis Model
The Chua hysteresis model belongs to the subset of state-space hysteresis models.
It is described in detail in [25, 26, 23] and an implementation of it is detailed in
[46].
The benefit of such a state-space model is that it is easy to implement for
real-time operation and is less computationally intensive than the Preisach model.
A further benefit of the Chua model is that its standard implementation captures
the phenomenon of loop widening, that is, the widening of the hysteresis loop with
increasing frequency. However, the simplicity of the Chua model comes at a cost: as
we will see later, it cannot model minor loops as accurately as the Preisach model.
The Chua model [23] can be described by the differential equation
d~)
(dt)
dv(t)
d)
The functions
f,
x
h(v(t)) x g [u(t) - f(v(t))] .
g, h, and w are the functions to be identified:
f
(3.32)
is the described
in the literature as the restoring function and g as the dissipation function. The
w function models the frequency dependence of the hysteresis and h is a weighting
function. In this thesis, h
=
1. Note that with the Chua model, there is no need to
store previous local minima and maxima.
131
3.4.1
Identification and Implementation of the Chua Model
Identifying
f
and g are straightforward. The function f can be determined from the
curve consisting of the midpoints between the ascending and descending branches of
the major hysteresis loop for equal values of B, where we have switched the input
and output variables u and v with H and B, respectively. Mathematically, we can
write
Ba(Hl) = Bd(H 2 ) =
f'(Hm),
(3.33)
where Hm is the mean value of H1 and H2 , expressed as
Hm- H, + H2
2
Graphically,
f
(3.34)
1 is illustrated in Figure 3-17.
B
H,,Bd(H,) ---------
H
HB-(H,)
H,
H,
---
f-l(H)
Figure 3-17: Identification of
dB
H
f for the Chua model.
'(H)
The function g is determined by the width of the major hysteresis loop. Mathematically, we can write
g9
1
(dB(t)) = d(t).
dt
(3.35)
Here d(t) is half the width of the major hysteresis loop at time t when B(t) is driven
with a cosine wave with zero offset. Graphically, g- 1 is illustrated in Figure 3-18.
The functions
f
and g can be implemented as lookup tables or as analytic functions
fit to the data.
132
B
dt,,
H Q,,B(t.)
dQ H(t,),B(t)
H(t,),B(1 1
)
d(t,
H
Figure 3-18: Identification of g for the Chua model.
The final function to determine is w. For frequency independence, we set w as
(dH'
H-
W
dt
_
1
dH
= 2irfOHmax I-d
(3.36)
'
W
where fo is the frequency of the cosine wave that was used to identify g. The parameter
Hmax is the maximum value of H reached with the cosine wave.
Figure 3-19 shows the major loop as well as minor loops constructed with the
frequency-independent Chua model.
Note that the major loop looks reasonable.
However, the minor loops show that the relationship between B and H is no longer
monotonic. This does not accurately model real materials. This is a major disadvantage of the Chua model compared to the Preisach model. This limitation is discussed
further in [27].
3.5
Summary
From a literature review, we learned that there are two general approaches to modeling hysteresis: physical modeling and phenomelogical modeling. We presented two
phenomelogical models for modeling ferromagnetic hysteresis. The first is the Preisach
model, an integral model that is capable of high accuracy. We learned that the graphical representation of the Preisach model provides insight into how the model works.
We presented two methods for implementing the Preisach model, the Della Torre
133
1.5
0.5-
0-
-0.5
-1
-1.5
-20
-10
0
H [A/m]
10
20
Figure 3-19: Major loop and minor loops constructed from the frequency-independent
Chua model.
implementation and the Hui implementation. We learned that both these methods
are both relatively straightforward to identify and yet are still capable of producing
realistic minor loops. With the Hui implementation, we developed a way to account
for asymmetry in the hysteresis loop. The second model we presented is the Chua
model, a state-space model that is simple to implement with a single first-order differential equation. We learned that the Chua model produces minor loops that do
not realistically model real materials.
In the next chapter, we develop a simulation model for the reluctance actuator
that incorporates the Preisach model. With this model, we investigate force errors
from hysteresis as well as other error sources.
134
Chapter 4
Reluctance Actuator Hysteresis
Modeling
In this chapter, we present a method for incorporating the hysteresis models presented
in the previous chapter into a full actuator model for simulation. This model can then
be used to predict and investigate actuator behavior. It can also be used as a design
tool to provide insight into how different actuator parameters affect results.
We first present the basic reluctance actuator model, where we implement Ampere's Law for the reluctance actuator and incorporate the Della Torre Preisach hysteresis model for the material B-H relationship.
We then use this model to investigate errors from hysteresis and gap disturbances.
We also develop theory for characterizing the errors from these sources. This results
in simplified formulas that can be used to approximate expected errors. We also show
that the predictions from theory match up well with simulated results.
Next we augment the reluctance actuator model with a flux feedback controller.
We develop theory for predicting the stiffness of the flux-controlled actuator with a
current drive and with a voltage drive. In the case of the current-driven compensated
actuator, we simulate the flux-controlled actuator model and show how the simulated
stiffness matches the theoretical stiffness. Finally, we present an overview of how one
might augment the basic actuator model in order to capture other phenomena and
to improve the model's accuracy.
135
4.1
Basic Reluctance Actuator Model
We begin with the lumped parameter model described by (2.7), rewritten here with
time dependence shown explicitly as
(4.1)
lFeH(t) + 2g()B( H(t)) = NI(t),
Ao
where we have replaced Bgl
with H(t), the magnetic field in the core material, and
where we have dropped the subscript from B9 . We write B as a function of H in
order to highlight the fact that it is a nonlinear function of H that includes hysteresis.
The difficulty in creating a model of the actuator that captures the hysteretic
behavior is that even if we have a model for the B-H relationship, we have no direct
access to H. We only have access to the input I. And because of the nonlinear
relationship between H and B, there is no straightforward way to solve for H explicitly
as a function of I.
Instead, we implement (4.1) in MATLAB Simulink and use a numeric solver to
solve it. In block diagram form, (4.1) can be represented as shown in Figure 4-1.
1 +
Input to actuator
(current)
B
H
B(H
)
NI
Fe
Hysteresis
rnodel
-
2g
P0 Fe
t
g
Gap disturbance
Figure 4-1: Block diagram for actuator model.
This gives us a model framework for simulating B(t) without having explicitly
to solve for H in terms of I or B. We will use this model to gain insight into the
actuator behavior. For real-time prediction, however, there is no guarantee that the
feedback loop in Figure 4-1 will be stable. We will address real-time prediction issues
in Chapter 6.
136
4.1.1
Incorporating the Hysteresis Model into the Actuator
Model
With the model framework described above, we can investigate the effect of magnetic
hysteresis in the reluctance actuator. To do so, we first need to choose an appropriate hysteresis model and replace the B(H) block in Figure 4-1 with this model.
We selected the Della Torre implementation of the Preisach model described in the
previous chapter. We chose this model because it can model minor loops well, unlike
the Chua model, and because it is relatively easy to implement compared to other
methods of implementing the Preisach model.
The output of the Della Torre model is dM/dH. To express the output in terms
of dB/dH rather than dM/dH, we note that B in the magnetic material can be
expressed as [16]
(4.2)
B = po(H + M).
Differentiating B with respect to H, we write
dBdB= pLO(1 + dH
dM )
(4.3)
Therefore, to obtain dB/dH from the Della Torre model, we add 1 to the output
dM/dH and multiply it by po. Since our Della Torre model output now is dB/dH
rather than B, however, we need to modify the actuator model to account for this.
To do this, we take the derivative of (4.1) as
dH
lFe
dt
dI
2Bdg
2gdB
+
+
=N-.
dt
to dt
to dt
(4.4)
We then use the chain rule to express dB/dt in terms of dB/dH as
'F
dH
edt
2gdBdH + 2Bdg
po dH dt
po dt
=
NdI
(4.5)
dt
The resulting block diagram implementing (4.5) with the Della Torre model is shown
in Figure 4-2.
This model takes dI/dt as its input rather than I. Integration is
137
then required to obtain B. For simulation, we took the desired current profile and
differentiated it numerically in pre-processing to obtain the desired dI/dt profile. This
dI/dt profile was then used as the input to the simulation model.
dH
dI
1
d
Fe
dB
+H
ddHdB
d
~-
d
dH
'
p
g
Delwa Torre
Model
dt
X
B
-rm-s
2g
Y 01Fe
dg
L2
IP 01Fe
dg
dt
Figure 4-2: Block diagram for actuator model with Della Torre hysteresis model.
4.1.2
Hysteresis Model Parameters
For simulation purposes, we chose to model a reluctance actuator prototype that we
designed for experiments in lab. The prototype actuator is a three-pole actuator with
nickel-iron laminated cut-cores. The actuator parameters and design details can be
found in Chapter 8.
The Della Torre parameters H, o-, Ad, H, and B, as defined in Section 3.3.2
(with B, replacing M,), were set to the values shown in Table 4.1.
Table 4.1: Della Torre model parameters
Parameter
Value
4 A/m
20.4
HC
a-
Hs
B8
Ad
I
104 A/m
1.215 T
1.807 T/ (A/m)
138
X
Here, H, was chosen to be the coercivity of the nickel-iron material as specified
by Magnetic Metals [57], the company that supplied the cut-cores for the actuator
prototype we designed. The saturation flux density was specified by Magnetic Metals
to be -1.5T.
We then chose B, = (0.9)2 x 1.5T = 1.215T.
The first factor of
0.9 reflects the lamination stacking factor, which is given by Magnetic metals as 0.9
for our lamination thickness of 0.004" (-100 pLm). The second 0.9 factor is a fudge
factor intended to reflect the fact that fringing fields in the air gap and leakage flux
will reduce the average flux density in the air gap. Since this model does not explicitly
model fringing fields or leakage flux, we chose to account for these effects on the force
by simply reducing the saturation flux density. The value for a was chosen to be a
value close to that found in the literature [9]. The variable H, was chosen to be a
large value so that the simulated H is guaranteed to remain between -H, and H,.
The value of Ad results directly from (3.18) as a consequence of setting the other
parameters. Figure 4-3 shows the major loop as generated by this hysteresis model.
B v. H
1.5
1
0.5-
-o
-
0-
-0.5
-1
-1.5
-20
-10
0
10
20
magnetic field [A/m]
Figure 4-3: Major loop of hysteresis model used for simulation.
Note that these values were chosen a priori, without any experimental verification.
For present purposes, this is acceptable since we are attempting to gain insight into
general behavior rather than accurately predict a particular actuator's performance.
139
4.1.3
Current Profile
The current profile used to drive the actuator simulation was chosen to approximate a
scaled version of an actual scanning profile. We started with a 2nd-order acceleration
setpoint profile with the following parameters: ama, = 225 m/S2,
smax = 4.5 x 106
m/S4.
jmax
= 22500 m/s3,
These refer to the maximum acceleration magnitude, maximum
jerk magnitude, and maximum snap magnitude, respectively. For a profile to be
2
"d-
order means that its first two derivatives are finite. A 2nd-order acceleration profile
corresponds to a 4th-order position profile.
The top plot in Figure 4-4 shows the
constructed acceleration profile.
Acceleration profile
400
7 0
-400
-
-200
0
0.1
0.3
0.2
0.4
0.5
0.4
0.5
Current profile
4
0
0
0.1
0.2
0.3
time [sec]
Figure 4-4: TOP: Acceleration profile for simulation. BOTTOM: Associated current
profile for simulation of single reluctance actuator.
Based on the acceleration profile, we then derived a feedforward force profile scaled
to the maximum force output of one prototype actuator. From this force profile, we
then derived a current profile for the reluctance actuator using (2.14). The bottom
plot in Figure 4-4 shows the current profile used for simulation. The current profile
shows pulses corresponding only to the positive acceleration pulses: this is because
140
we are simulating only one reluctance actuator, which can generate force in only one
direction. The opposing actuator is not simulated here.
4.2
Analysis of Errors from Hysteresis
We can use our simulation model to investigate the effects of hysteresis on accuracy.
There are two types of errors associated with hysteresis. The first is offset error: this
occurs when the current is driven to zero after the acceleration phase. Because of
hysteresis, the actuator core will not fully demagnetize in this state. If not compensated, this will result in a residual force and a residual negative stiffness acting on
the short-stroke stage during exposure, which can result in additional disturbances.
The second hysteresis error we are concerned with is the deviation in force between
the increasing portion and the decreasing portion of the current pulse. This deviation
will result in larger moving-average (MA) errors at the beginning of exposure if left
uncompensated. We denote this error simply as the hysteresis error, as distinct from
the offset error, although both are the result of hysteresis.
In this section we derive simple theoretical formulas that provide approximations
for the force error due to offset and to hysteresis.
These formulas are useful both
for designing a reluctance actuator to minimize hysteresis error and for predicting
expected order-of-magnitude errors from hysteresis. We then simulate the reluctance
actuator and show how the errors from simulation compare with the theoretical approximations.
4.2.1
Offset Error
In this analysis, we have assumed that the actuator was driven into or near saturation
before the drive current was returned to zero. When the actuator current returns
to zero after the actuator core has been magnetized, there remains a residual flux
density, ABff, in the core and air gap. This residual flux density can be thought of
as the 'remanence' of the sheared hysteresis curve. This is illustrated by point a in
Figure 4-5.
141
B
No air gap
With air gap
Figure 4-5: Hysteresis offset error AB,,f.
The residual flux density can be estimated from (2.30). We note that the core
magnetic field HFe corresponding to point a on the sheared hysteresis curve is located
at point b on the unsheared hysteresis curve. At this point, HFe
-H,.
We can
rewrite (2.30) as
-Hc + n
Ao
o
~Ha,
(4.6)
where we recall from Section 2.4.4 that n = 2g/1Fe and H,, = NI/1Fe- The applied
ABo,
~
ABoff
~~
n
Fe
2g
,
magnetic field H,, at point a is zero, so solving for AB,,f, we write
PoHe.
(4.7)
The offset error is proportional to the material coercivity and is inversely proportional
to the air gap. Therefore, choosing a core material with low coercivity and increasing
the actuator air gap will decrease the offset error.
An alternative way to derive this result is to consider the width of the B-I major
142
hysteresis loop at B = 0, which we denote as AIIB=o. From the definition of Ha, we
can write
AIIB=O
(4.8)
Fe AHaIB=O.
N
From Figure 4-5, we see that AHaIB=O= 2He, so we can express (4.8) as
Al =2lFeHc
Nle~
AIIBou-
(4.9)
This gives us the maximum width of the B-I hysteresis loop.
We now want to
determine the maximum height of the B-I loop, ABmx. Since the ascending and
descending branches of the B-I curve are approximately linear at the zero-flux crossto
ings, we can say that ABm, between the two branches will be approximately equal
the change in B that corresponds to AI = AIIB=o or AHa = AHaIB=o along either
the ascending curve or descending curve. We denote this change of B as AB' or
ABd, where the superscripts a and d refer to the ascending and descending branches,
respectively. See Figure 4-6 for an illustration. We can then approximate ABa from
the relation given by (2.8), where we substitute AIIB=o for I, shown here as
B
H
FB
Ama
Figure4-6:
eqa/oA"
isapproxmatel
A~a
143
A Bmax ~ AB" ~
goNAIB=o
2g
(4.10)
Substituting ( 4.9), we can write
ABmax -~
lOHe.
(4.11)
9
Noting that ABff is half
ABmax,
we arrive at the same result as (4.7).
We next want to determine the force error resulting from the offset (AFff). This
can be approximated in a straightforward manner by substituting ABoff for Bg in
(2.13), shown here as
AFOff
A .
(4.12)
2pvo
In terms of the material coercivity and actuator air gap, we can write
p)12 AH 2
AFO ff~
8F
8g2
"
(4.13)
We can express A in terms of the maximum F and maximum B, Fmax and Bmax,
respectively, by noting that Fmax
=
Blax/(2po)A. We write A as
(4.14)
A = 2pOFmax
B2
Bmax
If Bmax is close to the saturation flux density, B, we can substitute B, for Bmax and
then write AFOff as
AFoff
~
2F
BS,2
8g2
2
AFoff
~
plFefHc
If we stipulate a particular value for the
1Fe/g
Fmax.
(4.15)
ratio, this form allows us easily to
compare the expected offset errors among different materials for the same Fmax. Alternatively, if instead of stipulating a particular
144
lFe/g
ratio, we stipulate a particular
g, we can express
1Fe
as a function of Fmax and B, if we make some assumptions
about the actuator geometry.
Figure 4-7 shows the flux path on an actuator with
dimensions marked for calculating
1Fe-
pole face
depth d/l
Flux path
Figure 4-7: Actuator with flux path shown and dimensions marked.
lFe
2 dw
'Fe
2dw +
p
+ 2
2w + 2W.
1Fe
as
+
We can express the average flux path length in the core
(4.16)
We define the aspect ratios of the coil window and pole face as a = d/ww and
#
= dp/wp, respectively. The amp-turns NI is equal to JAw, where J is the current
density and Aw
=
wedw is the coil window area. We can therefore derive an expression
145
for dw in terms of these parameters as
wwd
=
Aw,
d2-w
= Aw,
dw
=
aNI ,N
aNImax
Jnax
-
(4.17)
where in the last line we have replaced NI and J with their maximum values, NImax
and Jmax, respectively. We can express NImax using (2.8) as NIma, = 2gB,/pto. If we
substitute this into (4.17), we can write dw as
dw =
2agB
p0max .m
(4.18)
Likewise, for ww, we can write
2gB
ap0 Jmax
(4.19)
We recognize that the center pole face area AP is half the total pole face area and
can be expressed using (4.14) as Ap = 1/2A = /oFmax/Bs.
146
Using this relation, we
can derive an expression for w, in terms of these parameters as
2w =
A,,
wP =
W
B
(4.20)
a
Substituting (4.18-4.20) into (4.16), we write
lFe =
2
=
2
lFe
C1
2agB,
pOJmax
2g
pOJm ax
VB +
+2
2gB8
apoJma
((a+1
V
B8
+2-
B +p2
1
pOFmax
B
#9
-Fmax
#a B8
(4.21)
where C1 and C2 are constants that depend on actuator geometry, air gap, and maximum current density. Crucially, C1 and C2 do not depend on specific dimensions, but
rather on the aspect ratio of the pole face and the aspect ratio of the coil window. In
this way, scaled versions of the same actuator for different materials can be compared
easily if we stipulate Jmax and g.
We can use (4.13) to predict the force offset error and compare this with simulation results. At a nominal gap of g = 500 pm, (4.13) predicts AFff = 0.0730 mN.
Simulation of the reluctance actuator using the current profile shown in Figure 4-4
results in AFOff = 0.0726mN, showing a very close match to theory. The simulated
force is shown in Figure 4-8. This error is only 2 x 10-5 percent of the full-scale force.
147
10-5
8
400
6-
300
-
W4
2
200
0
0
0.3
100
o'0
0.32
0.34
0.36
0.38
0.4
time [sec]
0.1
0.2
0.4
_0.3
time [sec]
Figure 4-8: Simulation of a reluctance actuator showing the offset force due to hysteresis.
4.2.2
Hysteresis Error
To derive an expression for the force deviation due to hysteresis, we first solve (2.2)
,
for B9 (dropping the subscript), where we have substituted Bg/pto for Hg, and Hg 2
and write
B =
IIl
-
2g
2g
(4.22)
HoNI
Fe-
We substitute this result into (2.13) to yield
F =
2po
pONI
potlFe
2g
2g
HFe
2po
4g2
+
_ /O 1Fe NIHFe
2g2
42
4g 2 HFe2
Now consider the force-current hysteresis loop shown in Figure 4-9.
(4.23)
We want to
determine AFh(I) = Fd(I) - Fa(I), the difference in force between the ascending
and descending curves for the same current value.
148
F
Fd
AF
I
Figure 4-9: AFh between the ascending and descending branches.
We can use (4.23) to write AF in terms of
AFh
--
/l
A
4g 2
2po
2g
2
+ 4g 2 (H e) _
Noting that Id = I
Fe NjI
2
H e
(4.24)
N 2(ia)2
0p2(gp
2
4g
H;e, and Hje as
+ 2
POFe(H)2
NIaHje _
= I and dropping the subscript on HFe for simplicity, we can
simplify this to
AFh =
A
2gFNI (Hd - Ha) + 42e ((Hd)2 - (Ha)2)]
2Fe2
4g2
I
Fe A
4g 2 2po
0 Fe
AFh
2 p~o
4g 2
A
2po
I H-
Ha) +
((Hd)2
_ (Ha)2)]
L Fe
I (Hd _Ha) + (H+d + Ha) (H-d
LleI (H
_-Ha) + 2t ( H d - H a
Ha)]
(4.25)
,
=N2( id)2
Ja, Id,
where in the last line, we have used the mean value of Ha and Hd, H
=
(Hd + Ha) /2.
Except at well into saturation and near I = 0, NI/IFe will dominate over H, so we
149
can approximate (4.25) as
p012 A '2N~
AFh
XF
Fe
8g 2
IFe
I (H
-I HaH
]
p0lFA I H(Hd-Ha .
P
_
(4.26)
We now have a simplified relation between the hysteretic force error on one hand and
the current and magnetic field on the other. For this relation to be useful in predicting
expected force deviation, Hd and Ha need to be related to I. This can be done for
instance by noting the relation between the sheared hysteresis curve and the material
hysteresis curve and using (2.30).
See Figure 4-10 for a graphical representation of
how Hd and Ha are related to I. For a particular value of I on the sheared hystersis
curve (dotted green curve), we note the value of B on the ascending branch, B'.
Using (2.30), we calculate the core magnetic field HFe on the ascending branch of
the blue curve by subtracting nBa/po from Ha = NI/Fe, recalling that n = 2 g/lFe.
This gives us Ha. Likewise, we can determine the Hd associated with I by noting the
value of B on the descending branch of the sheared curve, Bd, and subtracting the
quantity nBd /po from NI/Fe-
If the hysteresis loop is relatively square, we can further simplify the approximation in (4.26) by noting from Figure 4-10 that Hd - Ha . 2H, until very close to
saturation. Substituting this into (4.26) we write
POlFeANIH
2
AFh
2g
(4.27)
Using (2.8) and (2.13), we can express (4.27) in terms of B or F as
AFh
IFeABHe
FA B
g
IFeHe
l
g
2_A.
2u0 AF.
(4.28)
If we want quickly to compare actuators of different materials capable of the same
Fmax, we express A in terms of Fmax and B, using (4.14) with B, substituted for Bmax.
150
B
nd
HB d
/1!!0d,
~~73
HC
g
I'
Ba
NI/i
H
B/
Figure 4-10: A graphical representation of the relation between the B-I curve and
H,, and Hd.
We write
AFh
~
A Fh ra
where
1F,
~
lFeHe
2gB e,
4PIFmF
V/FmaxF,
(4.29)
can be expressed in terms of Fmax using (4.21) if desired.
Equations (4.27)-(4.29) allow us to estimate the expected hysteresis error in terms
of actuator parameters that can be approximated early in the design process. If we
know the maximum force required for a single actuator, we can estimate the maximum
hysteresis error by substituting Fmax for F in (4.29). This will give an overestimate
for the maximum hysteresis error, because the actual maximum will occur at some
value less than Fmax (intuitively, this makes sense, since we know that the hysteresis
loop must close eventually, so that AFh must start decreasing at some point before
Fmax). As with the offset error, choosing a material with low coercivity and increasing
151
the actuator air gap will decrease the hysteresis error.
The accuracy of the approximations given in (4.27-4.29) will degrade if the shape
of the hysteresis loop deviates significantly from the square shape because the range
for which it is valid to approximate H' - H' as 2H, is much smaller. In such cases,
if some knowledge of the shape of the hysteresis loop is known, then (4.26) can be
applied, which is a more general version of the hysteresis error approximation.
As was done with the offset error, we want to compare the hysteresis error predicted from theory with that predicted by simulation. Using (4.28) with F = Fmax, we
predict the maximum force hysteresis error, AFh,max, to be 0.65 N. Simulation results
in AFh,max = 0.56 N maximum force error. This is 0.15 percent of full-scale force. The
simplified formula shows reasonable agreement with simulation. Figure 4-11 shows
the maximum hysteresis error from the simulation.
I
The maximum error occurs at
2.92 A and F ~ 307 N, slightly lower than the 3.2 A and 367 N corresponding to
=
Imax
and Fmax used in (4.27) and (4.28), respectively.
F v. I
400
-
300 --
-
200
0
308
307.5
100 .
307
0
306.5
0
1
2
current [A]
AF
mx
~0.56 N
306
305
2.92
2.925
2.93
-- - - - - - - - c;ure tjA]
2.935
- - -- - -
-
305.5
Figure 4-11: Simulated force versus current showing the maximum force error from
hysteresis.
Figure 4-12 plots (4.27) and (4.28) for the second pulse in the current profile and
compares them with the simulated force error.
152
The approximations show a good
match with the simulated error except at high values of I and F.
A
Fh v.
A F v. F
I
0.7
0.8
0.6
0.6
simulation
-
Z
----
simulation
-
----
.
~~~~~0.5
~
theoretical approximation
--
theoretical approximation
-
0.7
.
--
--
--
0.4
0.4 --
0.3
0.20.2
0.3
--
0.1
0.1
0
0.5
1
1.5
2.5
2
3
--
0
3.5
300
200
100
400
force [N]
current [A]
Figure 4-12: LEFT: Comparison of AF from (4.27) with AFh from simulation.
RIGHT: Comparison of AFh from (4.28) with AFh from simulation.
4.3
Analysis of Error from Gap Disturbance
We can also use our simulation model to investigate the effects of a gap disturbance on
force accuracy. As was done in the previous section, we first derive a simple theoretical
formula for approximating the error from a gap disturbance. We then simulate the
gap disturbance and show how the simulation results compare with theory.
We start from the approximation given by (2.8), reproduced here and solved for
NI as
(4.30)
NI ~ -- B.
Po
We want to determine the change in flux density, ABd, resulting from a gap disturbance, Ag. To do this, we define two different states (denoted with the subscripts 1
and 2) of the actuator with identical currents but with a difference in gap between
them. We define I, = I2
=
Io, and we define g, = go and g2
the nominal gap. Likewise, we define B1 = BO and B2
153
=
Bo
=
9o
+ Ag, where go is
+ ABd, where Bo is the
air gap flux density at I = Io and g = go. We rewrite (4.30) for states 1 and 2 as
NI, ~
NI2g
Po
B1,
(4.31)
B2.
(4.32)
Po
If we subtract (4.31) from (4.32) and expand the terms for g2 and B2 , we can write
2
2
- (go + Ag) (Bo + ABd) goBo
Po
Yo
0,
goBo + BoAg + goABd+ AgABd - goBo
~
0,
BoAg + goABd + AgABd
~
0.
(4.33)
Assuming small perturbations, we can remove the second-order term AgABd and
solve for ABd as
ABd = --
Ag.
(4.34)
The change in flux density ABd is thus proportional the gap deviation and the nominal flux densily level and inversely proportional to the nominal gap.
Thus, gap
disturbances have a larger effect at high flux density levels.
A second way to derive the result in (4.34) is by substituting BO
+ ABd for B and
go + Ag for g directly into (2.8) as
B + ABd
poNI
2 (go + Ag)'
B + ABd
po NI/go
2 (1 + Ag/go)
(4.35)
Recognizing that Ag/go << 1, we can expand the left-hand side of (4.35) into a
154
Taylor series, truncating it after the first-order terms. We write
pONI
Bo + ABd O
(1
Ag
(4.36)
go
Noting that poNI/(2go) = Bo, we can solve (4.36) for ABd as
ABd'-
BoAg,
go
(4.37)
the same result we arrived at before.
We next want to determine the effect of the gap disturbance on force. We can
linearize (2.13) in terms of ABd around an operating point FO, Bo
F = F + AFd
-Lo"
2pLO
+ BoAAB
Yo
(4.38)
Noting that FO = B2A/(2po), we can solve for AF
B0 A
AFd
yo
AB.
(4.39)
Ag.
(4.40)
Substituting (4.34), we write
AFd ~B
P'ogo
Alternatively, expressing AFd in terms of FO, we write
2F0
2 Ag.
go
AFI
(4.41)
If we divide both sides of (4.41) by Ag and take the limit as Ag approaches zero, we
derive the stiffness of an uncompensated reluctance actuator, k,
kr
aF
-
sat
2 Fo
(4.42)
go
We see that the stiffness is negative and that its magnitude increases with force and
155
decreases with operating gap.
We can set an upper limit to the expected force disturbance by stipulating that
the maximum gap disturbance, Agmax, occurs at Fmax. Then
AF,max
-
2
(4.43)
Fmax Agmax.
go
For simulating the gap disturbance, we adapted a gap profile from data taken from the
long-stroke error on a lithography machine. The simulated gap disturbance is shown
in Figure 4-13. This gap disturbance is added to the nominal gap during simulation.
10
C)s
-
> -5
CL
CO
-10
0
0.1
0.2
time [sec]
0.3
0.4
Figure 4-13: Simulated gap disturbance.
Since the exact gap disturbance profile is unlikely to be known before designing
and implementing the reluctance actuator on a real stage, for comparing theory with
simulation, we replaced Agmax in (4.43) with an order-of-magnitude estimate of the
maximum gap disturbance rather than the exact maximum gap disturbance from
Figure 4-13.
In this case, we estimated Agmax as 10 pm.
The actual maximum
magnitude of the simulated gap disturbance is 9.2 pim.
Using this estimated Agmax, we approximated the maximum AFd to be 14.6 N.
The simulated force with gap disturbance is compared to the simulated force with no
gap disturbance in the top plot of Figure 4-14. Below it is a graph showing the force
error between the two cases. The maximum simulated AF is 13.8 N, showing good
156
agreement with theory. This error is 3.8 percent of full-scale force. Errors from gap
disturbance are thus expected to dominate over errors from hysteresis or offset.
Simulated force output
400
-
300
.2
200
-
-
-
-
no gap disturbance
gap disturbance
.-..-.-.-.-.-.-.-..---.
-.
-..
100
0.05
0
0.1
0.2
0.15
0.25
0.3
0.4
0.35
Force error from gap disturbance
20
10
-
-
-.-- .. ...
0
.... ..
. ...
0.05
0.1
0.15
0.25
0.2
time [sec]
0.3
0.35
.
0
-1 0..
0.4
Figure 4-14: TOP: Simulated actuator force output with and without gap disturbance;
BOTTOM: Simulated force error generated by gap disturbance.
4.4
Reluctance Actuator Model with Flux Feedback
We can use our model to investigate the effectiveness of flux feedback. Flux feedback
in block diagram form is represented in Figure 4-15. The block labeled 'Actuator
Model' represents the model developed in this chapter, B represents the actuator air
gap flux density, and Bm represents the measured air gap flux density.
Simulating a flux feedback loop with the nonlinear actuator model permits us to
investigate how well feedback control can linearize the actuator. We can explore the
effect of controller bandwidth on force accuracy and stiffness as well as the effect of
feedforward control. The sensor block represents the flux sensor. We can analyze the
effects of such things as sensor noise, sensor nonlinearities, and sensor offset on force
accuracy. We can compare and contrast different types of sensors. In the remainder
of this section, we investigate the effect of flux feedback on actuator stiffness. We
derive theoretical approximations for the compensated stiffness when the flux loop
157
Optional
feedforward path
F
FFB
B(Fd,g)
B
F(B,g
+u
oe
FluxC
controller
inversion
AtarBSeno
I
Flux feedback
loop
Figure 4-15: Block diagram of a reluctance actuator model with flux feedback.
includes a high-bandwidth inner current loop and when it does not.
For the flux
loop with current loop, we compare the theoretical approximation of the disturbance
rejection with simulation results.
4.4.1
Stiffness with Flux and Current Control
We first analyzed the expected change in flux density AB resulting from a gap disturbance Ag when the flux is being controlled with a feedback loop that includes a
high-bandwidth current loop. From this result, we then determined the expected stiffness AF/Ag. A block diagram of a linearized flux control loop with a gap disturbance
is shown in Figure 4-16.
The disturbance feedthrough transfer function D(s) can be approximated from
(4.34) as D(s) = -Bo/go, where Bo is the nominal flux density level and go is the
nominal air gap. The plant P(s) can be approximated as P(s) = poN/(2go), where we
have linearized (2.8) about the nominal operating gap. This plant takes current as its
input. Thus, this analysis ignores the electrical dynamics of the reluctance actuator.
Such an analysis is representative of a flux loop that includes a high-bandwidth inner
current loop (see Figure 2-17E).
The disturbance rejection transfer function B(s)/g(s) can be expressed as
B(s)
D(s)
g(s)
1 + C(s)P(s)
158
1
Bo
1 + LT(s) go
(4.44)
Ag
-
C(s)
-
P(s)
Disturbance
feedthrough
+
B
-
Bd +
D(s)
Plant
Controller
Figure 4-16: Block diagram of reluctance actuator flux control loop with gap disturbance.
where LT(s) = C(s)P(s) is the negative of the loop transmission. At frequencies
below the crossover frequency, LT(s) will dominate the denominator, and we can
approximate B(s)/g(s) as
B(s)
1
Bo
LT(s) go
g(s)
We can replace LT(s) with C(s)poN/(2go) and write
Bo
1
C(s)IpN go'
B(s)
g(s)
B(s)
_(s)
2Bo
r"'
(4.46)
C(s)poN'
With current and flux control, the flux loop disturbance rejection is independent
of gap. We also see that it is proportional to Bo, so the disturbance rejection will
deteriorate as the flux increases. Assuming that C(s) includes an integrator, the loop
disturbance rejection transfer function will increase with w at low frequencies.
We simulated a flux feedback loop using the model we developed earlier in this
4
chapter. Our controller was C(s) = 8.95 x 10 /s, which was designed to achieve an
approximate crossover frequency of 5kHz. To determine the simulated actuator disturbance rejection, we generated a frequency sweep of the model where the controller
159
maintained a constant reference Bo and the gap was given a sinusoidal disturbance
about the nominal operating gap. Figure 4-17 shows the simulated actuator gap disturbance rejection frequency responses for different levels of Bo when go
=
500 pm.
The simulations had some difficulties at some of the low frequencies, which accounts
for the discrepancies in the phase plots. Otherwise, the simulated frequency responses
confirm theory: the disturbance rejection magnitude increases with Bo and increases
linearly with w at low frequencies.
B(s)/g(s)
10-6
E
10
100
10
102
131
102
10 3
100
80
(D U
600
___
-
SB0
CL - 40
_
B0 =
0.2
=
0.5
B0=1.
202
100
10
10 4
frequency [Hz]
Figure 4-17: Simulated frequency responses of reluctance actuator gap disturbance
rejection with flux and current control for varying flux levels.
Figure 4-18 shows the simulated gap disturbance rejection frequency responses for
different nominal air gaps. As predicted by theory, the nominal gap has a negligible
effect on disturbance rejection.
In Figure 4-19, we compare a simulated gap disturbance rejection frequency response with the theoretical approximations given by (4.44) and (4.45). Both approximations match the simulated frequency response closely for frequencies below the
controller bandwidth.
From the disturbance rejection approximation, we can calculate the expected com160
B(s)/g(s)
1U
E
.E
10-6
E
10-8 01
2
CM
,'
3
10
-
-
-
--
- --
100-
10
102
10
100
80
60
4
-o
g 0 =400
ym
=500 lm
90 =600 ym
CL 200
100
103
102
101
104
frequency [Hz]
Figure 4-18: Simulated frequency responses of reluctance actuator gap disturbance
rejection with flux and current control for different nominal air gaps.
pensated actuator stiffness.
Expressing (4.45) in terms of AB and Ag and then
substituting into (4.39), we write
BoA
I
Bog
- LT(s)
Ao
B A
go
Ag
poLT(s)go
'
AF
2F
LT(s)go A'
(.7
where in the last line, we have used (2.13) to express the result in terms of F0 . The
expected stiffness of the compensated actuator, krc, is thus
kr =
rc
8F
2F0
- = -.F
LT(s)go(
g
(4.48)
By comparing this result with the stiffness for the uncompensated reluctance
161
B(s)/g(s)
10-2
E
L
0-4
E
1 0
10
101 1022
103
1 4
102
103
104
150
100
U)
50
simulation
-1/(1+LT)*(B 0/g
)
.c
-I/LT*(B
/1go)
0
100
101
frequency [Hz]
Figure 4-19: Simulated frequency response of reluctance actuator gap disturbance
rejection compared with theoretical approximations.
actuator, k, in (4.42), we see that kc is equal to
krc =
1
kr.
LT(s)
(4.49)
We can replace LT(s) in (4.48) with C(s)P(s), and express kc as
2 Fo
C 2)go'
uo
k_=-
4F0
.
poNC(s),
(4.50)
The compensated stiffness is proportional to the force level and independent of gap.
The stiffness will increase with w if the controller includes an integrator.
162
4.4.2
Stiffness with Flux Control and Voltage Drive
When we actuate the reluctance actuator with a voltage drive rather than with a
current drive, the flux dependency on gap changes.
we assumed a perfect current source as the input.
In the basic actuator model,
A more realistic model would
include a model of the electrical domain of the actuator as well.
We first derive
the uncompensated disturbance stiffness for the actuator when driven with a voltage
source. We then derive the stiffness for the voltage-driven actuator when compensated
with a flux controller.
Uncompensated Stiffness
Suppose we have a voltage drive, Vs, applied to the actuator coil terminals. Then
Kirchhoff's voltage law applied to the reluctance actuator is
=
RhdA
Vs = IR + d
dt
(4.51)
where I is the actuator current, R is the coil resistance, and A is the flux linked by the
actuator coil. The dA/dt term accounts for the actuator inductance and any speed
voltage from the changing gap.
The flux linked by the actuator coil is NM, where N is the number of coil turns
and <b is the magnetic flux through the center pole. If we ignore fringing and leakage
flux, we can write that <b = BAP, where AP is the center pole face area. Substituting
these relations into (4.51), we write
dB
Vs = IR + NA d.
(4.52)
If we solve (2.8) for I and substitute the result into (4.52), we can write
2gR
pON
dB
dt
163
(4.53)
Assuming a constant gap g 0 , we can derive the transfer function from V, to B as
B(s)
poN
2go
_
V(s)
1/R
,oAP,
2RgO
2
s
uoN
2go
+1
1
R
/R
+ 1s
(4.54)
('
where L = poAPN2 /(2go) is the actuator inductance. This is the open-loop transfer
function from voltage input to flux density output, so in the block diagram of Figure 4-
16, we set P(s) = B(s)/Vs(s).
To determine B(s)/g(s), we assume a state where g = go and B = Bo. We then
peturb g by an amount Ag, so that in the new state B = BO + AB and g = go + Ag
in (4.53). If V, is held constant between the two states, we can write
V
=
2(go+ Ag)R
PON
(Bo+-AB)+-NA,
d(B0 +AB)
d
dt
2R
2goRBo 2goR
2RB 0
=
B +
RAB+Ag+2
AgAB +NAPAb.
pioN
poN
ptoN
poN
(4.55)
Noting that the first term on the right-hand side is equal to V, and ignoring the
second-order term AgAB under the assumption of small perturbations, we can write
U=
2RB0
R .
2q
- 0 AB + Ag + NApzB.
poN
poN
(4.56)
Taking the Laplace transform of the above equation, we can solve for B(s)/g(s) as
[N
s--2goR~1
NAPS +
2RB
B(s)0
poN _poN
=
g(s),
-
B(s)
MON
g(s)
NAPs + 29 oR'
poN ~uN 2RBo
MO
2goR 1 ioApN 2 S-+
2goR
B(s)
B0
go
g(s)
164
1
+4
(s
Compared to the uncompensated disturbance rejection of a reluctance actuator
driven with a current (4.34), the voltage-driven uncompensated disturbance rejection
is low-pass filtered by the transfer function 1/((L/R)s + 1). A gap disturbance will
thus have less effect on the flux when the actuator is driven with a voltage source.
We can determine the uncompensated stiffness when the actuator is driven with a
voltage by expressing (4.57) in terms of AB and Ag and substituting this into (4.39).
This yields
AF
BoA
BO
ILO
go
B2A
1
1
1
s +1
Ag,
0L
AM,
Pogo S+1
2Fo
~g
1
(4.58)
+1 Ag.
go 7is
The stiffness is then
k, =
2F0
L
1
go Ls+1
.
(4.59)
Compensated Stiffness
To determine the compensated disturbance rejection when the flux loop does not
include a high-bandwidth inner current loop, we apply (4.44) using (4.57) for D(s).
We write
B(s)
D(s)
g(s)
I + C(s)P(s)
1
1 + LT(s)
1 Bo
1
LT(s) go Ls +(6
1
go -s +1J
Bo
(4.60)
where in the second line, we have approximated 1/(1 + LT(s)) as 1/LT(s), which is
valid for frequencies below the loop crossover frequency. We can replace LT(s) with
165
C(s)B(s)/V(s) and write
B(s)
1
Bo
g(s)
C(s) Bs
go
g(s)
s+iJ
Bo 1
go s+1
I/R
C(S) pO
2g0 Ls+i
B(s)
1
2RBo
poNC(s)
(4.61)
This result is similar to (4.46) for flux control with current loop.
Note that the
controllers C(s) will not be the same in the two cases. The disturbance rejection is
proportional to Bo and is independent of the operating gap go.
We can apply a similar analysis that we did for the flux loop with current control
to determine the compensated stiffness. We express (4.60) in terms of AB and Ag
AF
io)
iel
BoA(
L I
Po
LT(s)
)
yield
sta-:
ita- J-- ( A.C
)
a nSLu~bUbUUe lb into
anu
B BA
1oLT(s)go
2Fo
LT(s)go
B o-
1
0
s1
SR
Ag
AgI
Rs+1
(
1 )Ag.
(4.62)
The compensated stiffness kc is
2Fo
goLT(s)
166
1s+
R s+)1
(4.63)
Substituting C(s)B(s)/V(s) for LT(s) yields
~
~-
4.4.3
1
2F0
-
krc
=C(s)"2
T/go
(s+1)
4RFo .RF
poNC(s)
(4.64)
Comparison between Compensated Stiffness with and
without Current Control
It is useful to compare the compensated stiffness of the reluctance actuator with and
without a high-bandwidth current control loop. In order properly to compare the
two, we stipulate an identical flux loop crossover frequency for the two cases. In the
current-controlled case, we can achieve a high crossover frequency with an integral
controller K/s since the high-bandwidth current loop will have reduced most of the
phase loss in the plant. Our loop transmission can then be expressed as
LT(s)
-
To achieve the desired crossover frequency
N
s 2go
fc,
we set
(4.65)
|LT(s)
1 and substitute 27rfe
for s. We then solve for K as
K = 47rfgo
poN
(4.66)
Substituting K/s for C(s) in (4.50), we can write the compensated stiffness under
current control, kj, as
-
k
ke=
rie
4F0 poNs
cpN
47rfcgo'
- i
F0
S.
(4.67)
rfgo
For a flux loop with voltage drive, our loop transmission is C(s)B(s)/V,(s), which
167
we write as
LT(s) =C(s)
L/R
11R
poN
N
= C(s)
Is+1A
2go
.
(4.68)
Because of the pole from the inductance, we cannot use a simple integrator as the controller because there will be too much phase loss at the crossover frequency. Instead,
we use a proportional-integral (PI) controller of the form
C(s) = Ks
.
'7s
(4.69)
We can then write (4.68) as
LT(s) = K
rs+11
L/R
I
NA As + 1(
TS
.S1I
(4.70)
If the P1 zero is placed much lower than the crossover frequency, we can approximate
C(s) as K at f. To solve for K, we set JLT(s)I = 1 at s
K r27rfcj + 1
Kr27fej
1
NA
=
27rfcj. We write
LR
1
L27rfe
1
1
KNAP 27f,
(4.71)
The second line is a valid approximation if the PI zero and the pole from the inductance are both well below fc.
Solving for K, we write
K = 2rfNAP.
(4.72)
Our controller is therefore
C(s) = 27fcNAp S +.
TS
(4.73)
Substituting this result into (4.64), we can write the compensated stiffness with volt168
age drive, ke, as
kvrc
_
7s
_4RFo
(ON)
27rfcNA
Ts
7rfcp1_oAPN2 (-r
(Ts+ 1)'
)
(474)
+
kvrc
Noting that L = [poA N 2 /(2go), we can write kv as
-
rc
7rfcLgo (
s
The zero of the PI controller occurs at a frequency
the ratio of
f,
to
fa,
)
(4.75)
.
+
kv
fz
= 1/(27rT).
If we define a as
then we can write that r = a!/(27rfc). Then we can write kv as
( (4.76) fcs
RF
kv__o 2j-s 1)
At the low frequency limit (f <<
kv
k
fc/a),
we can approximate kr as
a R F0
c27r f, L 7rfgo'
=
rc
s
-
=-
=a
kv
27
k'k
27r fe L
fc
c
' k,e
c
(4.77)
where in the second line, we have substituted (4.50), and in the last line, we have
substituted the frequency of the inductive pole, fL = R/(27rL). A conservative value
for a is 10. For our actuator, the inductive pole occurs at approximately 5.5 Hz at a
nominal gap of 500 pm (see Chapter 8). If we select a crossover frequency of say, 5
kHz, then for these values, kv ~ 0.01kI,. Our low-frequency stiffness is a factor of
100 lower with a voltage drive compared to a current drive.
169
At the high frequency limit (f >> f/ca), we can approximate k' as
F0
kv rc= - rfcLgo
-R
The stiffness reaches a constant value at high frequencies.
(4.78)
This is in contrast to
the compensated stiffness when using a high-bandwidth current loop (4.67), where
the stiffness continues to increase linearly with frequency.
We thus see that one
advantage of not using a current loop in conjunction with the flux controller is that
we can achieve much lower stiffness at both low frequencies and high frequencies.
4.5
Augmenting the Reluctance Actuator Model
There are various ways in which we might augment the basic reluctance actuator
model in order to investigate other phenomena or to make the model more accurate.
In this section, we give a brief overview of some possible modifications to the model.
4.5.1
Angle Disturbance
The basic model permitted only a gap disturbance. If we model both halves of the
three-pole r-luctance actuator, we can simulate both gap disturbances and angle disturbances simultaneously and determine not only resulting force disturbances but also
torque disturbances. This phenomenon was discussed qualitatively in Section 2.6.2.
If we consider Figure 4-20, assuming small angles, we can write the gaps at each
pole face as
YL
. g-(L + w) 0,
(4.79)
gc
g - LO,
(4.80)
gR
g-(L - w) 0,
(4.81)
where g is the actuator gap without any angle disturbance, 0 is the angle of rotation
of the stage relative to the reluctance actuator stator and L is the distance from the
170
center of rotation to the center of the actuator target.
0
LL
.I Yr
(
BT
Flux path 2
Flux path 1
Figure 4-20: Reluctance actuator with angle disturbance.
For the left and right flux paths shown in Figure 4-20, we can write Ampere's Law
as
Flux path 1: NI
Flux path 2: NI
Bcgc
[to
Bcgc
[o
+ BL+L
+
HLlFe,
(4.82)
+BgR
[o
+ HR Fe-
(4.83)
Po
Here, HL and HR are lumped parameters representing the average magnetic fields in
the core and target material for path 1 and path 2, respectively. From Gauss's Law,
we can express BL and BR as material hysteresis functions of HL and HR, respectively.
Also using Gauss's Law, we can write Bc = (BL + BR) /2. With this in mind, we can
draw the block diagram of (4.82) and (4.83) as shown in Figure 4-21. The dotted blue
upper box encompasses the portion of the block diagram representing Ampere's Law
for flux path 1 and the dotted green lower box encompasses the portion of the block
diagram representing Ampere's Law for flux path 2. The coordinate transformation
block represents (4.79-4.81). The outputs of the model are BL, BC, and BR.
171
Flux Path I
+
BL
HLL
BL(HL)
Hysteresis
model
-
+
---
-
NI
HR
R
B
B(HR)
Hysteresis
-
model
R
2gRB
LI
C
Coordinate
Transformation
POl
g
0
71- - --------------------------------------------
Flux Path 2
Figure 4-21: Block diagram of a reluctance actuator with angle disturbance.
The force (F) and torque (T) output of the actuator can then be approximated as
F =
2po
r =
(B2 AL + B Ac + BR AR),
L
wA (B2- B2)
R
L
Apo
(4.84)
(4.85)
where Ap = 2AL = 2AR- In the case where 0 = 0, (4.84) reduces to (2.13) and (4.85)
reduces to zero.
4.5.2
Fringing Flux
As mentioned earlier, our basic reluctance actuator model does not account for fringing flux. Perhaps the simplest method for modeling fringing flux is to make the area
associated with the air gap reluctance a function of gap, reflecting the fact that the
flux will spread out more in the air gap as the air gap increases. The mapping between
172
this effective area, Aeff, could be determined from finite element analysis (FEA) for
example. We can apply (2.4) with A., replaced with Aeff(g) and solve for the air
gap flux density, Bg
B9
AFe
AeBf(g)
(4-86)
BFe,
where we replaced the subscript C in (2.4) with the subscript Fe for the sake of
generality. This gives us the average flux density in the air gap.
Fringing also affects the force generation: (2.13) does not account for fringing
One way to model the effect of
because it assumes that B, is perfectly uniform.
fringing on force generation is with a lookup table using average gap flux density
and gap as inputs, and with the force as the output.
This lookup table could be
obtained via FEA or with experimental results. A block diagram with fringing effects
accounted for is shown in Figure 4-22. The elements used to model the effect of
fringing are outlined in blue.
N -'Fe
1 + H
~
B(H)
Bee
B-
A
Hysteresis
B,
B F(Bg) F
AFe
g
model
g
Figure 4-22: Block diagram of a reluctance actuator with fringing fields modeled.
Another way to model the fringing fields is to apply the flux tube method discussed
in Chapter 7. By applying the magnetic circuit techniques analyzed there, we can
further augment the model to account for leakage flux. Finally, we can apply these
same techniques to discretize the single hysteretic lumped parameter core and target
reluctance into multiple hysteretic lumped parameters. We can do likewise with the
amp-turns in order to distribute the magnetomotive force spatially in a more realistic
manner. This permits us to represent local variations in the magnetic field within the
core and target and to explore which parts of the core and target saturate first.
173
4.5.3
Electrical Modeling
If we solve (4.52) for I, we can write
1 (dB\
I= -
Vs - NA
R (dt
-
.
(4.87)
We now have expressed the input of our actuator model (I) in terms of its output (B)
and Vs. We can implement (4.87) in block diagram form as shown in Figure 4-23.
The block labeled 'Actuator Model' represents the basic reluctance actuator model
developed earlier in this chapter.
1
VS +
Voltage
-
Actuator
Model
R
B
drive
dB
dt d
I dt
Figure 4-23: Block diagram of a reluctance actuator model with the electrical domain
modeled.
Modeling the electrical domain allows us to investigate the effects of having a
non-ideal current source. We can achieve the desired input current with a current
feedback controller and explore the effect of the actuator inductance and resistance
on the controlled current and output flux density and force. Alternatively, we can
plug this model into a flux feedback loop and investigate controlling the flux directly
sans current control.
4.5.4
Position Control
We can supplement the actuator model with a model of the actuated plant, i.e., the
short-stroke stage. This allows us to investigate the actuator force accuracy required
to maintain the required stage position accuracy, and to investigate the effect position
feedback control has on actuator force accuracy.
174
The simplest model would include a single actuator pair that can actuate a singleDoF stage.
A more comprehensive model could include four reluctance actuators
acting on a 2-DoF stage (y and 6). This schematic is shown in Figure 4-24. Each
actuator block has two outputs, a force (F) and torque (r). The variables
Od
9d
and
are the gap disturbance and angle disturbance from the long-stroke stage tracking
error. The position controller and any flux controllers are not shown in the figure.
Actuator
9d
Stage
Model
y
Od
Actuator
Model
Model
9d
9d
F
F2
Actuator
Model
0
Od
Actuator
F3
F
Model
Od
9d
Od
Figure 4-24: A 2-DoF model of the short-stroke stage with four reluctance actuator
models and gap and angle disturbances.
Additional augmentations to the stage model can be easily envisioned. For example, we could add a model of the long-stroke stage, or we could include additional
degrees-of-freedom to the model.
4.6
Summary
In this chapter, we have developed a reluctance actuator model that incorporates
ferromagnetic hysteresis and air gap disturbances. We have developed simple formulas
that allow the designer to estimate expected force errors from hysteresis and gap
disturbances. We have also listed some ways in which the model could be improved.
Table 4.2 lists the formulas for approximating errors from offset, hysteresis, and
gap disturbance.
The formulas are expressed as functions of force rather than as
functions of flux density or current. The reason for this is that the required force is
more likely to be known in advance than either the required actuator flux density or
current, which are typically determined later in the design process. For the lab pro-
175
__0__
totype actuator that we simulated, the maximum errors as percentages of maximum
force are also listed for each error source. Both the theoretical and simulated maximum error percentages are listed. We learned that gap disturbances will generate the
largest errors for the uncompensated reluctance actuator.
Table 4.2: Error sources for a reluctance actuator
2B Fmax 2.0 x 10-
2.0 x 10-5
Offset
Max error as %
of Fmax (theory)
Max error as
of Fmax (sim)
%
Theoretical
approximation
Error source
AF__ =
Hysteresis
AFh
Gap disturbance
AFd
-
FmaxF
2pOI
--
Ag
0.18
0.15
3.68
3.77
In Table 4.3, we list formulas for approximating the reluctance actuator stiffness
for several scenarios. These scenarios include an uncompensated actuator driven with
a current source, an uncompensated actuator driven with a voltage source, an actuator compensated with a flux controller and driven with a current source (e.g., the
flux loop includes an inner high-bandwidth current loop), and an actuator compensated with a flux controller and driven with a voltage source. Note that CI(s) and
Cv(s) refer to the different controller transfer functions required for a current-driven
actuator and a voltage-driven actuator, respectively.
We discovered that for flux
loop bandwidths that would be typical for our application (i.e., several kHz), the
flux-controlled actuator with voltage drive is likely to result in a much lower stiffness
than a flux-controlled actuator with current drive.
The reason for this is that in
the voltage-driven case, the effect of a gap disturbance will be further filtered by the
actuator inductive pole.
176
Table 4.3: Reluctance actuator stiffnesses
Controller topology
Theoretical approximation
Uncompensated, current drive
k'
Uncompensated, voltage drive
k'
Flux control, current drive
kc
Fc
voltage drive
Flux control,
___________________________
2F
=
=-oNC(s)4F
=v
177
T-8
4RF
ioNCv (s)
178
Chapter 5
Eddy Current and Power
Dissipation Modeling
In this chapter, we augment the basic reluctance actuator model presented in Chapter 4 to account for eddy currents. We can use this augmented model to determine
the effect of eddy currents on force generation and to select appropriate values for
key design parameters, such as lamination thickness.
We begin by presenting the theory behind eddy current generation.
We distin-
guish between two sources of eddy currents: classical eddy currents and excess eddy
currents. We then develop lumped parameter models for approximating the average magnetic fields produced by these eddy current sources. These models are valid
for frequencies well below the skin frequency of the material.
We develop theory
to approximate force errors generated by eddy currents and compare these errors to
those obtained from simulation with the augmented actuator model. Finally, we investigate the power dissipated from hysteresis and from eddy currents. We develop
approximate formulas for predicting the dissipated power and compare the results
with simulation.
179
5.1
Diffusion Equation
Eddy currents arise in conducting material owing to the changing magnetic flux in
the actuator core and target. These eddy currents result in an internal magnetic field
that opposes the applied magnetic field, thus reducing the overall magnetic flux in the
material. This phenomenon can be derived from Faraday's Law (2.32) and Ampere's
Law (2.1). Applying these equations and Ohm's Law, we derive the diffusion equation
[39] as
V 2 H =e-- -
(5.1)
t
where ae is the electrical conductivity of the ferromagnetic material. If we can solve
the diffusion equation for H or approximate it, we can add these terms to (2.2) and
then determine the effect of eddy currents on the air gap flux density.
Solving (5.1) explicitly is often not possible. Instead, we employ a concept known
as loss separation as presented in more detail in [14].
Loss separation takes into
account the fact that for a wide range of ferromagnetic behavior, the total loss in
the material can be decomposed into the sum of three contributions: hysteresis loss,
classical eddy current loss, and excess eddy current loss. We can often approximate
a lumped-parameter value of H associated with each contribution.
Since we have
already derived a lumped-parameter model of the material hysteresis, we focus our
efforts here on classical eddy currents and excess eddy currents.
Before discussing
classical and excess eddy currents, however, we give a brief overview of the concepts
of skin frequency and skin depth.
5.2
Skin Frequency and Skin Depth
As part of the design iteration process, calculating the expected skin frequency and
skin depth aids in choosing the core material and lamination thickness.
The skin
depth is a measure of how far an applied magnetic field will penetrate a ferromagnetic
material at a given frequency [39]. It is defined as the distance from the surface of the
conductive material at which the flux density has decreased to 1/e of its surface value.
180
Alternatively, for a given thickness of material, the skin frequency is the frequency at
which the skin depth is half the material thickness, thus indicating that the magnetic
field no longer fully penetrates the material at this frequency.
For a lamination with thin rectangular shape, the skin frequency, f8, can be derived
from the 1-D diffusion equation and is given by [56]
4
fs =
r
d2
,
(5.2)
where pr is the relative permeability of the material and d is the lamination thickness.
The maximum relative permeability Pr of NiFe is listed by Magnetic Metals as
about 60,000. We use the maximum value to determine a worst-case skin frequency.
The conductivity of NiFe is listed by Magnetice Metals as 2.22 x 106 S/m, and d for
our prototype actuator is 100 pm. Using these values, we calculate
f,
to be 736 Hz,
well above the main frequency content of the force profile.
The skin depth, 6, for a given frequency,
f,
is given by [39]
j= 2
(5.3)
VlrIIO~ef
We can approximate the maximum frequency, fmax, in our simulated B-profile as
fmax ~ max
(dB
1
Sdt )27rBma,
.
(5.4)
For our profile, the maximum dB/dt is 112 T/s and Bmax = 1.12 T. This results in
an f m,,ax of about 16 Hz. Substituting this into (5.3), we calculate 6 = 2.2 mm, over 20
times greater than the lamination thickness. These calculations suggest that classical
eddy currents should have only a small effect at the frequencies of interest in our
prototype actuator.
181
5.3
Classical Eddy Currents
The classical eddy current loss is the loss associated with the scale of the actuator
geometry.
It is derived from the diffusion equation applied to the ferromagnetic
material assuming that it is perfectly homogeneous and consists of a single magnetic
domain. In this section we follow the analysis presented in [14] in deriving the field
associated with classical eddy currents.
If the frequency of the drive signal is well below the skin frequency of a core
lamination, then the applied magnetic field will almost fully penetrate the material
and we can approximate the internal magnetic field as being uniform within the
material.
Figure 5-1 demonstrates this graphically in one dimension for a single
lamination. The variable Ha is the applied magnetic field and H(x) is the internal
magnetic field as a function of x. The variable H is the spatial average of H(x). The
spatial dimension x is defined as equal to zero at the center of the lamination.
H(x)
d
H
HHH
Figure 5-1: Magnetic field distribution within a single lamination (left) and approximation with uniform H (right).
For the 1-D case, (5.1) simplifies to
02 H
8
S Ue
.
(5.5)
Because we are approximating the internal B as spatially uniform, we can integrate
182
CX2
(5.5) and write
aH
Ox
OH
Ox
-
f dB
J edt
dB
a,dt x +C1(t),
dt
~
(5.6)
where P represents the spatial average of the internal B and Ci(t) is the constant of
integration. We can integrate a second time to solve for H(t) as
H(t)
H(t)
dx
o, d!3 x +C(,
~
Sdt
'I dt2 +C(~+2t,(5.7)
dt 2
where C2 (t) is a second constant of integration. If we assume no skin current, then
H(x) = Ha at the boundaries x = d/2 and x = -d/2.
We can substitute these values
of x into (5.7) and write
x =:
d
2
H(t) =Oi
d
x
2. H(t) =
d~d2
d
2
dt 8
d~d2
d8
ed
+C1(t)
d
(t),
(5.8)
C1 (t)d +C 2 (t).
(5.9)
2
+C
2
d
If we subtract (5.9) from (5.8), we find that C1 (t) = 0. Using C1 = 0 in (5.8) we solve
for
C2 (t) as
C2 (t) = H(t)
-Ore
df d 2
Ue 7
(5.10)
8
Substituting the results for C1 (t) and C2 (t) into (5.7), we write
(5.11)
H(t) ~0 Ha(t)
2 dB
4
We want now to determine the spatial average of H(t), PI(t). We do this by taking
183
the x-average of (5.11). We arrive at
dT3
.
12 dt
Ued2
Ha(tt)
(t) -
(5.12)
The last term in (5.12) is the average magnetic field resulting from classical eddy
currents. We denote this as He, and we write
He(t) =e 7-ed
12
2
dB
(5.13)
dt
where we have replaced the B notation with B for simplicity.
We can see from
this equation how the lamination thickness can have a large effect on reducing eddy
currents since Hec increases as d2 . We can replace the magnetic field HFe in (2.2) with
HFe = Hh + HeI, where Hh is the field associated with hysteresis and is approximated
The modified version of (2.2) is therefore
by the average internal magnetic field f.
(Hh + Hd) lFe + Hg 1g -+H9 2 g
=
NI,
(5.14)
or
HhlFe +
ed2 dB
+ -B
Po
12 dt 1
= NI.
(5.15)
We can then modify our basic actuator model with an additional fccdback path that
represents the classical eddy current term. This is shown in Figure 5-2, where the new
feedback path is represented in blue. For the sake of simplicity, we have represented
the hysteresis model as having the output of B directly rather than dB/dH as was
done in Figure 4-2.
As was done with hysteresis and gap disturbances, we would like to estimate the
expected error resulting from classical eddy currents. We define state 1 as the actuator
model without classical eddy current effects (2.2) and state 2 as the actuator model
with classical eddy current effects (5.15).
For identical NI for the two models, we
define B1 = B and B 2 = B + ABei, where ABc1 is the the deviation in flux density
184
NI
B(H)B
H
Hysteresis
model
1F-
2g
dB
aid2
12
HC1
dt
d
dt
Figure 5-2: Block diagram for actuator model including the classical eddy current
term.
resulting from classical eddy currents. We can write the two models as
Hh1lFe+-gB
= NI
(5.16)
2g
= NI.
(5.17)
/to
Hh2lFe +
Jed2 dB
2
12 dt
Fe+
-(B + ABj)
YO
If we assume that Hj is small relative to Ha, then H1 will not have a large effect on
Hh, and Hhi ~ Hh2. If we subtract (5.16) from (5.17), we can solve for ABd as
pcAO
edl2Fe
dB(
dt
24g
To determine the corresponding deviation in force, AFj, we can write
AFd=F
2
-F1
2 _Fj
=,
=
-
B2A
2po
A
(B + AB")2 - B2]
A
2BABc + (ABd)2]
21to
AF
~
B A
2po
A
-BAB,
/po
185
(5.19)
where in the last line, we removed the second-order term: (ABs1 ) 2 . Substituting
(5.18), we derive
Aed2 lFeA dB
4Fg B--.
24g
dt
(5.20)
We can write AFl in terms of I, where we approximate dB/dt using (2.8) and assume
that g is constant,
OU-dFeAN
F
96g 3
AF e
- Yoe
dI
Id(
(5.21)
Finally, in order to express (5.20) in terms of F, we first solve (2.13) for B
2oF
B
1
A
.
(5.22)
We then differentiate this equation with respect to t and express dB/dt as
dB
p
dt
for F -7 0.
dF
'
2AF dt
Substituting (5.22) and (5.23) into (5.20) and restricting ourselves to
single-sided operation (B > 0), we obtain
A Fcd
AFd
_
ed2lFeA
2pa0F
24g
A
iooed2 le dF
24g
dt
po dF
2AF dt'
(5.24)
Force deviations from classical eddy currents will increase with lamination thickness and force slew rate (dF/dt) and inversely with air gap. Figure 5-3 shows the
theoretical approximation of force deviation from (5.21) compared to the deviation
shown by simulation. For the simulation we used the current profile from Figure 4-4
as the input to a Simulink implementation of the augmented model of Figure 5-2.
The scaled force profile is also shown. The match between the theoretical approximation and simulation is very close. The maximum force deviation predicted by (5.21) is
8.80 mN and the maximum force deviation predicted from simulation is 8.86 mN. This
amounts to 0.0024 percent of full-scale force, indicating that classical eddy currents
186
are expected to have negligible effect for this particular actuator design.
Force deviation due to classical eddy currents
0.01
0.005
0
-0.005
-0.01
0
0.1
theoretical approximation
simulation
scaled force profile
0.4
0.3
0.2
time [sec]
Figure 5-3: Comparison of AFj from (5.21) with AF1 from simulation. Red line is
a scaled force profile to provide a time reference.
Note that the maximum force deviation from eddy currents occurs during the high
dF/dt portions of the force profile, as predicted by (5.24). Another feature to note is
the small offset that can be observed during portions of zero dF/dt in the simulated
force deviation. This is the result of numerical inaccuracies in the simulation solver:
as the simulation time step is decreased, this offset is reduced.
We would like to determine whether our assumption that the internal H can
be approximated as uniform is reasonable.
In Figure 5-4, we plot the simulated
applied magnetic field, Ha, the simulated average internal magnetic field, Hh, and the
simulated average classical eddy current field, Hd. Hh shows only a small deviation
from Ha and Hd << Ha, verifying our assumption.
5.3.1
Effect of Gap Disturbances on Classical Eddy Currents
Suppose we are driving the reluctance actuator with a known current profile (e.g.,
suppose that no flux feedback is being used). We can augment (5.21) to account for
gap disturbances. In this case, (5.19) becomes
AF~-
pI-ed2 lFeAN 2
(
96g
187
dI
I dt
dt
I2 dg
g
g dt
.
(5.25)
6
H
a
-Hh
4
H1
-2
E
-4
-6
0
0.1
0.2
0.3
0.4
time [sec]
.
Figure 5-4: Simulated Ha, Hh, and He1
If Ag << go, the nominal gap, we can approximate g as go and write
PF2O'ed2 lFeAN 2
I
dI _
jdt
96g
go
I
2
dg
-1
dtJ
(5.26)
(
AFc, ~-d
While the gap disturbance profile in a lithography application is unlikely to be known
ahead of time, if the expected range of dg/dt can be estimated a priori, we can
postulate an upper bound for (5.26).
Suppose we estimate that the maximum dg/dt will be approximately 10 mm/s.
This is roughly the maximum dg/dt from the gap disturbance profile shown in Figure 4-13. For our simulated current profile, the I- dI/dt term has a maximum value of
512 A 2 /s. The second term, assuming that I and dg/dt are uncorrelated, has an upper
bound of 198 A 2 /s. The first term dominates, but the second term could contribute
nearly 40% additional error. It should also be noted, however, that max(I - dI/dt)
does not necessarily occur at the same time as max(I), so that the second term will
not reach its maximum at the same time that the first does.
5.4
Excess Eddy Currents
If the ferromagnetic material under consideration were perfectly homogeneous and
consisted of only a single magnetic domain, classical eddy currents would be the end
188
of the story as far as eddy current phenomena are concerned.
In the real world,
however, materials are not perfectly homogeneous and consist of multiple magnetic
domains. This gives rise to additional eddy currents known as excess eddy currents
[14].
Excess eddy currents result from domain walls undergoing motion from the
externally applied field.
Because the magnetic domain structure of a material is often very complex and
not known precisely, it is typically difficult to model excess eddy currents from first
principles.
Nevertheless, Bertotti in [14] has done an admirable job of deriving a
plausible model for excess eddy currents that fits a wide variety of materials. His
final result [14, p. 425] for the excess field, HeeC, is
Hexc(t) =sgn
(dB
-dt
noVo
~
(0eGA
1 +-
2
dB(.
nn~ 0
dt
dt
-
1),
(5.27)
where no and V are phenomenological parameters that characterize the material. The
parameter G is a dimensionless constant approximately equal to 0.1356. The variable
A, is the cross-sectional area of a single lamination. The details of this derivation can
be found in [14]. Here as with the derivation for H1, Hexc is assumed to be small
relative to Ha, such that the internal magnetic field can be approximated as uniform.
The parameters no and V depend on the microstructure of the material and are
related to features such as grain size and the number of correlation regions1 . Bertotti
provides some theoretical equations for approximating Vo and no. Normally, however,
these values are found by empirical curve fitting.
We can augment (5.14) to include Hec as
(Hh + Hd + Hexc) lFe + H9 1g + Hg2g
=
NI.
(5.28)
'Correlation regions describe the number of internal degrees of freedom in the magnetization
process of a material. Bertotti [14] describes it thus: "Certain aspects of the magnetization process
turn out to be describable by a small number of independent degrees of freedom, each degree of
freedom corresponding to the magnetic flux linked to some correlation region of the material."
[emphasis in the original]
189
Substituting (5.13) and (5.27) for He1 and Hexc, respectively, we write
HhlFe +
aed 2 dB
e
12 dt
Fe
dB
dt
+ sgn
noVo
B~~no~o(
2
4creGAi dB
rV 0 yd1lt
n[o
1
2g
-
po
B
NI.
(5.29)
We can include an additional feedback path in Figure 5-2 to account for the excess
eddy currents. The modified block diagram in shown in Figure 5-5. Both eddy current
NI
H,3H
_+
e
e
tHysteresis
)
paths (classical and excess) are shown in blue.
model
P 0Fe
H,
ca
od 2
9
dB
2
Ht
dntd
Figure 5-5: Block diagram for actuator model including excess eddy current term.
Of interest is to explore the low- and high-frequency limits on the behavior of
Hexc. For the low-frequency limit, i.e., dB/dt << (nfV0)/(4aGAi), we can expand
He, using a Taylor series applied to a function in the form of (1 + x) 1 / 2 for x << 1,
where in this case x = 4oeGAi/(noVO)|dB/dt. We write
~~sgn
Hex
noV (
2
14eGAI dB
2 noV0
dt
ii
'
HdB
dt
exc~sgn(1+
dB noVo (14-eGAi dB
2
Sdt
2
2 noVo
dt
eGA dB
no
dt
(5.30)
In the last line, we have used the fact that sgn(x)Ix| = x. For the high-frequency
limit, i.e., dB/dt >> (noVo)/(4-eGAi), the second term underneath the square root
190
will dominate the other terms, so we can write
Hexc - sgn
dB VF
dB
-eV0GAi d
\dt
dt
(5.31)
We see then that for low frequencies, like the classical eddy current field, the excess
eddy current field increases linearly with frequency. At higher frequencies, however,
the excess eddy current field follows a square root dependency on frequency. We
see also that excess eddy currents are much less dependent on lamination thickness,
d, than classical eddy currents are. Classical eddy currents vary with the square of
lamination thickness. However, with excess eddy currents, A, is proportional to d,
not d 2 . Therefore, at low frequencies, the excess eddy current field increases with d,
and at high frequencies, it increases with v d. Decreasing lamination thickness will
still help reduce excess eddy currents, but not to the same extent that it will with
classical eddy currents.
To predict the force errors expected from the excess field, we follow a similar
procedure to that used for the classical field.
We first derive the change in flux
density due to excess eddy currents, ABexc, from (5.29) as
0
(5.32)
ABexc
-sgn
dB
--
POlFenoV
4-eGAi dB
4g
-
Following the analysis used to derive (5.19) and applying it to the excess field, we
can write
AFexc
A
- BABexc.
191
(5.33)
Substituting (5.32), we write
AFexc
-B
g
dt) -4g
AlFernoVoB
dt )dB
A-g
B
-sgn0
rld
sn
I
(eG0jnBo dBdt
POlFen0oVo
-sgdB
SAB
1+4eGAi
noV 0
B
dt
(5.34)
We can express (5.34) in terms of I for a constant gap by substituting (2.8) and its
derivative for B and dB/dt, respectively,
( pN dI\ AlFenOV p10NI
~2 - sgn dI1 1 0
4g
2g
(2g dt
(V
poAlFeNOVO
~ r- sgn ( d
dII
2
+ 2pocGAiN dI - 1)
dt
noVog
)
8g
(5.35)
(
AFeXC
1+ 4-eGAi p1oN dI
n+Vo
2g dt
.
)
AFexc
Expressed in terms of F for F z 0 by substituting (5.22) and (5.23) into (5.34), we
write
AFexc
AFexc
~
-sgn
( dF)
-sgn
dt
AlFeoVO
2A10F
4g
A:
iFeno V
4g
2 110AF
C
1o dF
2AF dt
1+ 4a;GAI
n 0 Vo
(/1
+ 4aGA
0Vo
110
2AF
dF -1
.
7Ht
(5.36)
Expressing A in terms of Fmax and B, using (4.14) with B, in place of Bmax, we write
dF
Sdt
lFenOV
4p 2FmaxF
4g
B2
-sgn
AFexc
(dF POlFenOV
- sgn
-s dt ) 2gB8
"I1
+4'-eGA
n V
dF
B
4FmaxF dt
(
FmaxF
+2uGABs
no
maxF
IdFI
1)
.
(
)
AFexc
(5.37)
For simulation purposes, we selected values of no = 125 and V = 0.472 A/m.
192
These values were adapted from values documented in [14] for a silicon-iron specimen,
accounting for the effect of differences in lamination cross-sectional area. These values
are used to get a general idea of the behavior of excess eddy currents. For accurate
prediction, experimental verification of no and V would be required.
Figure 5-6 shows the theoretical approximation of force deviation from both classical eddy currents and excess eddy currents compared to the deviation shown by
simulation. The scaled force profile is also shown. The match between the theoretical
approximation and simulation is very close. The maximum force deviation predicted
by theory is 30.3 mN and the maximum force deviation predicted from simulation is
30.9 mN. This amounts to 0.008 percent of full-scale force. Excess eddy currents contribute about 22 mN to this error. For this particular actuator geometry and for the
chosen values of no and Vo, excess eddy currents play a larger role than classical eddy
currents for the force profile simulated. However, the overall effect is still minuscule
compared to errors from gap disturbances and hysteresis.
Force deviation due to classical and excess eddy currents
0.04
0.02
0
-0.02
-
-0.04
-
0
0.1
0.2
time [sec]
Figure 5-6: Comparison of AFddy = AFd
from simulation.
theoretical approximation
simulation
scaled force profile
0.3
0.4
+AFexc from (5.21) and (5.35) with AFeddy
Figure 5-7 shows the simulated applied magnetic field, H, the simulated average
internal magnetic field, Hh, the simulated average classical eddy current field, Hd, and
the simulated average excess eddy current field, He. We can see that Hexc << Ha,
again demonstrating that approximating the internal field as uniform is a reasonable
assumption.
193
6
H
Ha
Hd
4
-_
2
H1
- He
Hexc
0
-2
C~c
E
-4
-6
0
0.1
0.2
0.3
0.4
time [sec]
Figure 5-7: Simulated Ha, Hh, He1 , and Hexc using (5.27) for Hexc.
Figure 5-8 illustrates the loop widening that results from eddy currents for one
pulse of the simulated force profile. The inner loop is the B-Ha curve when no eddy
currents are present. The middle loop is the B-Ha curve when accounting for only
classical eddy currents. The outer loop is the B-Ha curve when accounting for both
classical and excess eddy currents. The figure shows that the presence of eddy currents
requires a larger applied field in order to reach a particular flux density. The 'kinks'
in the two outer loops are the result of the shape of the force profile. Specifically, it
is caused by the transition from the zero-snap phase to the non-zero constant snap
phase of the force profile, which results in 'kinks' in the dB/dt profile.
Because of the linearizing effect that comes from shearing the material B-H curve,
this loop widening of the material B-H curve has a much reduced effect when viewed
from the perspective of the B-I domain. Figure 5-9 shows the loop widening in the
B-I domain.
5.4.1
Excess Eddy Currents in 50%Ni-50%Fe Laminations
It was later discovered that while a wide variety of ferromagnetic materials follow the
square-root dependency for Hexe given by (5.27), this is not true of 50%Ni-50%Fe.
Instead, the field associated with excess eddy currents in nickel-iron have been found
194
B v. Ha
1.2
1
E 0.8
0.6
0.4
---
0.2
0
-6
.
-4
no eddy currents
classical eddy currents
classical and excess eddy currents
6
4
2
0
-2
applied magnetic field [A/m]
Figure 5-8: Loop widening from eddy currents for the simulated B-Ha curve.
B v.
I
1. 2
8
E 0.
0.6
0.4
-
0..2 -
--
0
0
1
no eddy currents
classiclI eddy currents
classical a d excess eddy currents
2
current [A]
.565
0 .564
3
0 .563
.562
'.561
1.595
1.6
1.605
current [A]
Figure 5-9: Loop widening from eddy currents for simulated B-I curve.
to follow a relationship given by [13]
HeX
UeGAi
1 +~
1+
4
dB
dt
(5.38)
d0
Since a quintic equation is not generally solvable, we apply numeric methods instead
to solve for Hexc. Our lab prototype actuator has nearly the same lamination cross-
195
sectional area as the specimen investigated in [13], so we use the same V determined
there, Vo = 3.8 A/m. One advantage of (5.38) is that only one parameter needs to be
determined.
For simulation, we did not use the augmented model shown in Figure 5-5 because
the numeric solver had difficulty solving the algebraic loop with (5.38) in place of
(5.27), even with time delays inserted. Instead, we used the real-time model described
in Section 6.6.
Figure 5-10 shows the simulated deviation in force due to both classical eddy
currents and excess eddy currents, with the excess field described by (5.38).
The
maximum force deviation predicted from simulation is 0.426 N. This amounts to 0.12
percent of full-scale force. We see then that excess eddy currents play a much larger
role in nickel-iron than was originally predicted with the square-root dependence given
by (5.27). This agrees with the findings in [13], where the 50 Hz losses due to excess
eddy currents took up a much larger proportion of the total losses compared to most
other materials tested.
Force deviation due to classical and excess eddy currents
0.5
0
-simulated
-0.5
0
0.1
Figure 5-10: Simulation of AFeddy
currents.
0.2
time [sec]
force error
scaled force profile
0.3
0.4
AFc1 + AFxc using (5.38) for the excess eddy
If we plot the simulated Hee along with Ha in Figure 5-11, we see that the peak
He., is on the same order of magnitude as the peak Ha. Approximating the internal
field as uniform is therefore not a justified assumption in this case, so we should take
the simulation results for AFddU in Figure 5-10 with a grain of salt. This also suggests
196
that NiFe is not the best core material to use in applications where minimizing eddy
current effects is important.
15
a
N
Hh
--
E
5
H1
H
-
10
0
-5E
-10
-15
0
0.1
0.2
0.3
0.4
time [sec]
Figure 5-11: Simulated Ha, Hh, He1 , and Hexc using (5.38) for Hexc.
5.5
Power Dissipation
Power is dissipated within the ferromagnetic core and target from hysteresis losses and
eddy current losses. This dissipated power will cause heat generation in the stator and
target. There are multiple drawbacks to heat generation. Ferromagnetic properties
have some dependence on temperature, so a calibration done at one temperature will
not necessarily be accurate at another temperature. Hall sensors have a temperature
dependency, so if the pole face is heating up, this will affect a Hall sensor measurement.
Temperature rises also cause the stator and target to expand: this can cause an
unmeasured gap change and can also cause distortion to the chuck.
We would thus like to be able to estimate the expected magnitude of the power
dissipated from hysteresis and eddy currents.
These estimates can be used in a
thermal model to estimate expected temperature rise and thermal distortion. Making
use of the loss separation concept mentioned earlier, we treat hysteresis loss, classical
eddy current loss, and excess eddy current loss separately. This allows us to write
197
the total internal power loss, P, as
P =
where
Ph
+Pc + Pexc,
(5.39)
is the loss associated with hysteresis, Pc is the loss associated with classical
eddy currents, and Pex is the loss associated with excess eddy currents.
For many materials, the total loss per cycle can be expressed as a function of
frequency,
f, as
[14, 70]
P
-
f
= CO +C1f +C2
,
(5.40)
where Co, C1, and C2 are constants. The hysteresis loss per cycle is constant, while
the classical eddy current loss per cycle varies linearly with frequency. The excess
eddy current loss per cycle often follows a square-root dependency on frequency.
The energy per volume, e, in a magnetic body is given by [14, 70]
e = JHdB.
(5.41)
The energy lost while traversing a B-H loop is thus given by the area enclosed by
the loop. The power dissipated per unit volume, p, can then be written as
p= H
5.5.1
dB
dt
.
(5.42)
Hysteresis Loss
Figure 5-12 shows a generic acceleration profile for a scanning stage. The duration of
a single pulse is T and the period of the entire scan is Te.
If we consider a single pulse, we can estimate the energy per volume lost owing to
hysteresis (eh) as
eh
~ 2HcBmax,
(5.43)
where Hc is the coercive field and Bmn, is the maximum flux density reached. This
approximation is shown graphically in Figure 5-13. The area enclosed by the hystere-
198
a
6
-I
Figure 5-12: A generic acceleration profile for a scanning lithography stage.
sis loop (shown in blue) is approximated with a rectangle of width 2H, and height
Bmax (shown in green). This approximation will improve the more square the material
hysteresis loop is.
B.
B
H1
-H,
H
Figure 5-13: A rectangle of width 2H, and height Bmax approximates the energy per
volume lost from hysteresis.
The average hysteretic power per volume dissipated during the pulse is then
2
(Ph)p
HcBmax
T
(5.44)
The () denotes the average power per volume and the subscript p denotes that we
are referring to the average power of a single pulse.
If we consider the entire scan cycle, then the duty cycle is defined as D = 2T/T,
199
i.e., the fraction of the entire scan during which the stage is accelerating. The average
power per volume dissipated over the entire scan for a single actuator is
1
(Ph)
(Ph)
d
(Ph)p-D,
2DHBmax
T
~
(5.45)
The reason for the factor of 1/2 in the first line is that since a reluctance actuator
can only pull, any single actuator will only be on during one acceleration pulse per
scan cycle rather than during both.
We can express (5.45) in terms of Fmax
2DHe
(Ph)
2toFmax
A
.
T
(5.46)
.
We adapt (5.45) to our simulated profile in Figure 4-4 and predict (Ph) to be 45 W/m3
To determine the actual simulated power per volume dissipated from hysteresis, we
use
dB
(Ph) = (Hh
),
(5.47)
which results in a value of 42.6 W/m3 , a close match to the theoretical approximation.
5.5.2
Classical Eddy Current Loss
Based on (5.42), the instantaneous power loss per volume from classical eddy currents,
Pci, is Ha - dB/dt. Substituting (5.13), we can write
PcI
=ed2
= 12
dB
dt
2
.
(5.48)
Or in terms of F for F $ 0, we can write
poaed2
Pc
24AF
dF
2
200
Jed2 B2 (dF)
48FmaxF
K
2
.
(5.49)
Suppose that the exact force or flux profile is unknown ahead of time, but we
know the approximate peak force or peak flux and the pulse width, and we want to
estimate the average power per volume dissipated from classical eddy currents. We
can estimate the force pulse with its first Fourier component (assuming a square wave
for the force pulse, which should bias the result toward a conservative one, i.e., an
overestimate). The average power dissipated per volume for a sine wave of amplitude
Bo and frequency
f
is given by [14]
2
(Pcl~sin =
6
Ced2 B f 2 ,
(5.50)
where the sin subscript denotes that we are taking the average power per volume for
a sine wave.
We note that for a square wave with amplitude Bmax, the fundamental component
will have an amplitude of 4/7r - Bmax. We can substitute this for BO into (5.50) to
derive an approximate value for the average power dissipated per volume for a single
pulse
8ed
(PdOp
The frequency
f
2
B
3j~
2
max
f 2.
of the fundamental component has a value of
(5.51)
f
= 1/(2T). Multi-
plying (5.51) by D/2, we derive an approximation for the average power per volume
dissipated from classical eddy currents over the entire scan cycle as
(Pd)
Dae2d2
3T
B2Wax.
(5.52)
In terms of Fmax, we can write
P
(C~
d2
2Dpooe
2
AT
3 AT2 Fmax.
(5.53)
3
.
Applying this formula to our simulated force profile, we predict (pd) to be 1.83 W/m
To calculate the exact value (pd) for simulation, we use
dB
(Pd) = (H dB ).
201
(5.54)
This results in 1.99 W/m 3 , again a close match to the theoretical approximation.
5.5.3
Excess Eddy Current Loss
The instantaneous power loss per volume from excess eddy currents, Pexc, is Hexc
dB/dt. If Hexc can be expressed by (5.27), then we can express Pexc as
dB
noVo
pe:(t)
dt
2
2
4ceGAi dB 3
Y+no
+
dt
dt
0
.
(5.55)
Assuming that the low-frequency content will not contribute significantly to power
loss because of the dependency on dB/dt, we can use the high-frequency limit of Hexc
in (5.31) to approximate pexc(t) as
_ _ dB
GO
Pexc (t) ~ \/VeGAV
0 d
dt
3/2
(5.56)
.
In terms of F for F # 0, we write
3/4
Pexc(t) e
\fUeGAiV
dF
2AF
3/2
.
dt
(5.57)
If the force or flux profile is not known exactly ahead of time, using (5.56) or
(5.57) to approximate the average power per volume dissipated during a scan is not
as straightforward as was done for (pci) because of the (dB/dt)3 / 2 term.
Suppose
though that we have an idea what the maximum dB/dt ((dB/dt)max) will be and
approximate the jerk phase of the acceleration profile as having a constant dB/dt
equal to this maximum. We can then write the average power dissipation per volume
for a single acceleration pulse as
(Pexc)
2amadB
2
jmaxT
3/2
(5.58)
eGAiV 0 (dB
d
max
Here 2amax/(jmaxT) is approximately the fraction of the acceleration pulse with non-
zero jerk. We can further approximate (dB/dt)max as Bmaxjmax/amax. Making this
202
substitution, we write
2
(Pexc)p
T
(5.59)
Bm ax
maxeGAt
Jmax
Finally, for the average power dissipated per volume for the entire scan cycle, we
multiply by D/2
D
T-
(Pex7)
jmaxueGAiVo
3
/2
(5.60)
Bmax.
Or, in terms of Fma, we write
D
T-
T
jmaxweGAV0
m
amax
2poFma
A
.
(64
(5.61)
)
(Pexc)
With this formula, we predict (pexc) to be 7.7 W/m 3 for our simulated profile. To
calculate the exact value (pci) we use
(Pexc) = (Hexc
).
(5.62)
This results in 6.6 W/m 3 , a reasonably close match to the theoretical approximation.
It should be noted that a number of approximations have gone into estimating power dissipation from eddy currents, particularly in regards to the excess eddy
current calculation.
The numbers calculated here should not be expected to have
high-precision predictive power and may become less accurate with different force
profiles. However, the order of magnitude of these numbers gives us confidence that
power dissipation due to eddy currents should be negligible compared to that from
hysteresis and from resistive loss in the coils.
5.6
Summary
In this chapter, we have augmented the basic reluctance actuator model introduced in
Chapter 4 to include eddy current effects. We have presented models for classical eddy
currents and for excess eddy currents valid for frequencies below the skin frequency.
203
From these models, we have developed formulas for approximating force errors due to
eddy currents. These formulas are listed in Table 5.1. We learned that for our profile,
excess eddy currents are likely to dominate over classical eddy currents. However,
both of these are expected to contribute negligible force errors compared to errors
from hysteresis or from gap disturbances. We also learned that excess eddy currents
in 50%-50% NiFe follow a different law from that followed by excess eddy currents in
many other ferromagnetic materials.
Table 5.1: Errors from eddy currents in a reluctance actuator
Theoretical
approximation
Error source
Classical eddy currents
4F
IFedEte
\(cft
Excess eddy currents
AFexc
-
sgn
(d
2gB 5
FmaxF
1
20yeGAiBs
IdF
fl 2eVoV~maF
We also developed formulas for approximating the average power per volume dissipated in the reluctance actuator core and target during scanning. This dissipated
power was comprised of contributions from hysteresis loss, classical eddy current loss,
and excess eddy current loss. Table 5.2 list the approximate formula for each contribution. Also listed are the theoretical and simulated numeric values for the average
power dissipation per volume for the simulated prototype actuator. We learned that
hysteresis loss will dominate eddy current loss for our application. However, we expect the both the hysteresis loss and eddy current loss to be small compared to the
resistive loss in the coil.
In the next chapter we investigate methods for implementing the actuator model
for real-time flux estimation and control.
These real-time techniques can also be
used as alternate methods for simulating the results presented in this chapter and the
previous chapter.
204
Table 5.2: Reluctance actuator ferromagnetic power dissipation sources
Theoretical
approximation
Power
dissipation source
Hysteresis loss
(Ph)
2DH Bmax
Classical loss
(Pc)=
2
Do, d
B2
D-
"r^axe
Excess loss
(PeT)
205
Bmax
Simulated
Theoretical
value [W/m 3]
value [W/m 3 ]
45.0
42.6
1.8
2.0
7.7
6.6
206
Chapter 6
Reluctance Actuator Modeling for
Real-Time Flux Estimation
The actuator model presented in Chapter 4 cannot be used for real-time estimation
as presented because it includes an algebraic loop. If time delays are added to break
the algebraic loop, there is still no guarantee that the loop will be stable in real-time.
Thus, real-time model implementation is a significant challenge.
In this chapter,
we present several model implementations for real-time estimation that incorporate
hysteresis and gap disturbances. We discuss advantages and drawbacks of each.
Among the different implementations presented in this chapter are two novel techniques for estimating the actuator flux. The first, introduced in Section 6.4, is to use
an observer with an adaptive controller gain. The observer permits us to estimate flux
from the actuator current and gap measurements without having explicitly to solve
for the magnetic field in the actuator core. The adaptive controller gain linearizes the
observer loop gain even with the presence of the highly nonlinear material hysteresis
model within the loop. This adaptation performs a similar function to feedback linearization. This feature allows us to maintain accurate flux estimation over the entire
B-H plane.
The second novel technique, introduced in Section 6.6, is to introduce a change
of variables into the reluctance actuator model that allows us to model a sheared
version of the B-H hysteresis relationship. This shearing makes the plant model seen
207
by the observer much less nonlinear. The observer structure with sheared hysteresis
model (SHM) is shown to be much more robust to large gap disturbances than the
observer using the original unsheared hysteresis model. We present simulation results
for each implementation. For a couple implementations, we also present preliminary
experimental results. Finally, we present a conceptual analysis of the error behavior
of the different observer implementations for real-time flux estimation.
Except for the implementation discussed in Section 6.7, which uses the Chua
model, all the real-time flux estimation schemes presented in this chapter use the
Preisach model for modeling the hysteresis. However, the methods presented in this
chapter are all general enough that other hysteresis models could be used in place of
the Preisach model.
6.1
Motivation
As discussed in Section 2.5, one way to linearize the reluctance actuator is by feedback
control of the actuator flux (see Figures 2-16 and 2-17). In this thesis, we use a sense
coil as the flux feedback sensor. However, this requires that we have a way to estimate
the actuator flux at low frequencies since a sense coil is AC-coupled. Developing a realtime model that takes as its inputs the actuator current and air gap can provide this
flux estimate without requiring the use of any additional feedback sensors. Additional
details on the overall feedback scheme using this low-frequency model in tandem with
a sense coil can be found in Chapter 10. In this chapter, we focus only on developing
the real-time model for the low-frequency flux estimate.
6.2
Original Model with Stabilizing Filter
If the original model used for simulation in Chapter 4 and shown in Figure 4-1 is
stable or can be made stable, then it may be possible to use this model for real-time
estimation. We first fit the Della Torre hysteresis model to experimental data. We
then analyze the loop transmission of the resulting estimation loop for stability and
208
find it wanting. We then add a filter to stabilize the loop and simulate the resulting
model reliably.
6.2.1
Hysteresis Model Identification
To determine the parameters for the hysteresis model, we used experimental data from
the reluctance actuator prototype on the air bearing testbed described in Chapter 8
and then fit the Della Torre Preisach hysteresis model to it. Figure 6-1 shows an
experimental B-I curve taken from the reluctance actuator at a nominal gap of g =
500 pm and driven with a 10 Hz sine wave. The current was measured using a sense
resistor tied to the low end of the actuator coil and the flux linkage derivative was
obtained from a measurement taken from a sense coil wound around the center pole
face. The sense coil measurement was numerically integrated in post-processing to
obtain an estimate of the flux density. The target was attached to the testbed motion
500 pm using position feedback on the motion
stage. The gap was maintained at g
stage. See Chapter 10 for details.
B v. I
1.5
1
-0.5
-3
-experimental
-
near approximation
..
-2
-1
0
1
2
3
current [A]
Figure 6-1: Experimental B-I curve and its linear approximation for an actuator gap
of 500 pm.
A fit to the linear portion of the B-I curve is also shown. The slope m of this
line is 0.47 T/A. From (2.8), we can replace poN/(2g) with m. We can use (2.30) to
determine the material B-H curve for the actuator. We rewrite (2.30) in terms of I
209
and solve for HFe, the core material magnetic field as
HFe =
NI
lFe
2gB
6.1)
/-0 1Fe
or
N (
1
I
J.
B
m
lFe
(6.2)
/
HFe =
Using this equation, we can solve for HFe from the experimental B-I data and plot
the B-H curve as shown in Figure 6-2. We fit the Della Torre model introduced in
Chapter 3 to this data. We set H, = 120 A/m,
add a linear part to the model, dB/dHi1 ,
=
- = 20.4, and B, = 1.2 T. We also
250po, to improve the fit. The output of
the hysteresis model is therefore dB/dH = dB/dHDT + dB/dHn, where dB/dHDT
is the output of the unmodified Della Torre model. Figure 6-2 shows the fit of the
Della Torre model to the experimental major loop. The fit is imperfect and could be
improved, but for our present purposes of developing an operational real-time flux
estimator, it is sufficient.
B v. H
1.5
reversals in H
--.-.
0 - -.
-0.5
-1
-
-400
-200
0
experimental
Della Torre fit
200
400
magnetic field [A/m]
Figure 6-2: B-H major loop derived from experimental data and Della Torre model
fit.
We note that the experimental data shows that B is not monotonic with H. This
is indicated by the red circles in the figure. This is believed to be the result of the
position controller not maintaining a perfectly constant air gap. The measured gap
210
data is shown in Figure 6-3. The air gap shows a peak fluctuation of approximately
4p1m about the nominal.
We attempted to compensate for this in the B data
plotted in Figure 6-2 by multiplying the integrated sense coil data by g/go, so that
the compensated flux density, Bc, is equal to B - g/go. This compensation is based
on (2.8), where B (x I/g. We can express B, at g = go as
,u0NI
O
BC =
(6.3)
2go
Likewise, we can express B in the general case as
B=
2g
(6.4)
.I
If we solve for I in both equations and set them equal to each other, we can write
2goBe
MON
2gB
pzON
(6.5)
If we then solve for Bc, we can write
Be
90
=
Bl.(6.6)
Therefore, to determine the B-I relationship at a particular constant gap from data
in which the gap is changing, we multiply the measured B by g/go. This B, variable
is what is plotted in Figure 6-2. This is an improvement upon the uncompensated
B, which is plotted in Figure 6-4.
The compensated Be-H data from Figure 6-2 is likely still imperfect because the
approximations in (6.3) and (6.4) do not take into account saturation. They assume
a perfect inverse relationship between B and g throughout the entire range of B. In
Figure 6-5, we plot the experimental B-H data from an actuator that was clamped
with 500 pm plastic shims between the stator and target.
Clamping the actuator
avoids the air gap fluctuation that occurs using the air bearing setup. The clamped
actuator setup can be seen in Figure 6-18.
211
The B-H data looks much improved,
Gap v. time
...-...-
-.
504
502
.
.-..
~
~
...~
.~
CO 00
CD
RnA
-
.- .... -...
498
0
0.05
0.1
time [sec]
0.15
0.2
Figure 6-3: Air gap data for experimental B-H data from Figure 6-2.
B v. H
1.5,
-.... -...
1
-..
2(1
'0
X
-
E 0.5F
...
-..
0
..
-. . .... -.. ..
-1
-
-0.5-
0
-200
0
200
400
magnetic field [A/m]
Figure 6-4: B-H experimental major loop with no compensation for changing gap.
indicating that the changing air gap is indeed the culprit for the non-monotonic
relationship between B and H in Figure 6-2.
This procedure for finding the B-H curve highlights one of the drawbacks of this
method of estimating flux: since we do not have direct access to H in the steel, we
have to construct H from the original B-I experimental data.
212
B v. H
1.5
0
- 0 .5
.
...
.-.
.-...
-...
-...
-..
--.
-1
00
-200
200
0
400
magnetic field [A/m]
Figure 6-5: B-H major loop derived from experimental data and Della Torre model
fit.
6.2.2
Analysis of Loop Stability
To determine whether the loop illustrated by the block diagram of Figure 4-1 is stable
or not, we need to estimate the loop gain and phase shift. The loop gain, ILTI, is
B
|LT|= H
2g
(6.7)
POlFe'
where IB/HI is the gain of the B(H) block in Figure 4-1 at any relevant operating
point.
The 2g/(PolFe) block will not contribute any phase loss to the loop, so the loop
return ratio phase, ILT, is
LLT = LB/H,
(6.8)
where LB/H is the phase of the B(H) block at any relevant operating point. Therefore, determining the loop gain and phase comes down to estimating the gain and
phase of the hysteresis block.
Since hysteresis is a nonlinear phenomenon, we must linearize the hysteresis operator in some way in order to estimate its gain and phase. Here we opt to use a
describing function analysis to do so [72, Chapter 6].
We begin by approximating the hysteresis operator as simple non-ideal relay with
213
a lower output value of -Bmax
thresholds of H, and -Hc.
and an upper value of Bmax and input switching
See Figure 6-6.
B
Bmax
-Hc
HCH
H
-B max
4
Figure 6-6: A simple relay as an approximation for the hysteresis operator.
We denote the describing function for such a relay as BD(H), which is given by
[72] as
BD(H)
4Bmax
rHO
Z - sin--1
(
Ho
(6.9)
where HO is the amplitude of the assumed sinusoidal input signal. As an initial test,
we would like to see if our real-time flux estimator can recreate the experimental
B-I curve shown in Figure 6-1.
Therefore, we set Bmax in (6.9) to the maximum
experimental B, Bmax = 1.3 T, and HO to the maximum H, HO = 360 A/m, both
taken from Figure 6-2. These values result in a gain for BD(H) of 0.0046 T - m/A
and a phase of -19.5'.
Note, however, that the gain and phase are both highly
dependent on the input signal amplitude.
We can approximate the loop transmission as
LT- LT
=-
B
H
2ga/B/H,
BH
pOlFe
4BmaxN
in
7rHO miFe
27.4Z - 19.50,
Ho
214
(6.10)
where in the second line we have substituted N/m for 2g/po based on (6.2).
In addition to the phase loss from hysteresis, we will have phase loss from the
time delay associated with sampling. For example, we will have an additional -90'
of phase loss at a frequency of f,/4, where
f,
is our sampling frequency. For stability,
the loop gain must be less than 1 when the total loop phase reaches -180' . Since our
loop gain is constant at 27.4 at all frequencies and there is no defined loop crossover
frequency, we expect our model to be unstable for real-time operation.
6.2.3
Stabilizing Filter Design
We can add a low-pass filter in the loop to stabilize it. This low-pass filter must meet
two criteria: 1) for stability, it must reduce the loop gain to below 1 before the loop
phase reaches -180'; 2) it must have a low-frequency gain of 1 so as not to affect
the model accuracy at low frequencies. Note the contrast between this filter and a
controller in a typical feedback loop: in a typical feedback loop, we want a high gain
on the controller filter within the loop bandwidth, while here we ideally want a gain
of 1 within the loop bandwidth.
The reason for this difference is that in a typical
feedback loop we want the output to track the input, which requires a high controller
gain, whereas in our loop, we want to keep the loop as unaffected as possible while
at the same time stabilizing it.
Figure 6-7 shows the model with a stabilizing low-pass filter in the loop. The
variable Be is the estimated flux density after the filter.
Our new loop transmission, denoted as LT2 , now includes the stabilizing filter.
The new loop gain magnitude can be expressed as
B
LT2 1 LT
=ILTIF(s)|=
(S)
H
N
M1Fe
|F(s)|
,
(11)
where F(s) is the stabilizing filter transfer function. The phase of the loop transmis-
sion, denoted by
OLT 2
, is
@LT 2
=
OBH
+
215
OF
+
OT,
(6-12)
Mel
B
H
_
-
current
Be
Stabilizing
Filter
N/Ife
Hysteresis
Estimated
fluxdensity
Model
B1
B2g/(muO*lfe)
[t'gd]
gap
2g(mu*fe)
Figure 6-7: Reluctance actuator model with stabilizing filter for real-time.
where
#BH
is the phase associated with the hysteresis model,
stabilizing filter, and
#T
OF
is the phase of the
is the phase of the time delay from sampling. The phase
margin, 0m, is defined as 0,
frequency, the frequency where
=
180 +
at
qLT2
f
-
fc,
f,
where
is the crossover
ILT2 1 = 1. A positive phase margin is required for
stability.
Suppose
#T
f,
=
20 kHz. Then, if we set our crossover frequency,
will contribute 360 of phase loss at
fc.
to be at 2 kHz,
If our stabilizing filter is a one-pole filter,
this will add an additional 90' of phase loss at crossover. With
-19.5
fe,
#BH
estimated as
, this gives us a nominal phase margin of 0m = 35'.
It remains to design F(s) so that the loop gain is 1 at
f
=
fc.
If F(s) is a one-pole
filter with DC gain of 1, we can write it as
F(s) =
1
Ts + 1
The gain of F(s) must be equal to 1ALT at
of F(s) as a function of
f
.
(6.13)
f = f, to force crossover at fc. The gain
for frequencies well above its corner frequency is ~1/(27rfT).
Solving for r gives
|JL=
LT|. (6.14)
27rf,
This results in r = 2.2 ms. This corresponds to a corner frequency of 73 Hz. We
216
implemented this filter in the loop and simulated the resulting model. However, we
discovered the loop to be unstable.
The reason posited for this is that the gain
associated with the hysteresis model,
|B/HI,
is dynamic:
it changes substantially
depending on where on the hysteresis loop one is. The maximum occurs near H = H,
and is 0.142 T - m/A.
This is 30 times greater than the gain obtained from the
describing function analysis. The upshot is that the original loop gain will at times
be much larger than that indicated by (6.10).
This will result in a much higher
crossover frequency and danger of instability.
A new filter design was based on this maximum gain of JB/HI = dB/dHmax,
resulting in a much more conservative design.
The maximum loop gain, ILTImax,
without the filter is
Using this value for
ILTI
N
dB
= dB
ILTIm
a
= 850.
(6.15)
in (6.14), we calculate a value of 0.0675 s for -r. This corre-
sponds to a corner frequency of 2.35 Hz for F(s). From this result, we can anticipate
that achieving an accurate flux estimate will be difficult since the corner frequency
is so low. This means that our stabilizing filter will affect our loop even at very low
frequencies.
The simulated estimated B-I curve using this stabilizing filter is shown on the
left plot of Figure 6-8, where the input was a 1 Hz sine wave. We have included the
experimental B-I curve from Figure 6-1 for reference. We can see that even at the
low frequency of 1 Hz, the estimated B shows significant distortion. This distortion is
even more apparent when plotting the simulated B-H curve, which is shown on the
right plot.
One thing to note is that the distortion is greater near saturation. The reason for
this can be understood by considering the model from the 'closed-loop' perspective.
If we were to assume a linear model, the closed-loop transfer function from I to B,
is given by
Be(s)
I(s)
_
N
G(s)F(s)
1Fe 1 + G(s)H(s)F(s)'
where G(s) represents the linear transfer function approximation of the hysteresis
217
B
v. I
B v.
1
--
experimental
-
simulation
E 0.5-
E 0.5
(0
0
-3
H
1.5
1.5
-2
-1
0
1
current [A]
2
3
_-40
-200
0
200
magnetic field [A/m]
400
Figure 6-8: LEFT: Simulated B-1 curve from Figure 6-7 model compared with experimental B-I curve; RIGHT: Corresponding simulated B-H curve.
model and H(s) is the feedback gain N/(mlFe). When G(s) is very large, e.g. near
H = He, and we are operating at a frequency below 1/(27r7), then G(s)H(s)F(s) will
dominate in the denominator, and so F(s) in the numerator will cancel with F(s) in
the denominator, and the distortion from F(s) will be negligible.
On the other hand, when G(s) is very small, for instance near saturation, then
the 1 in the denominator will dominate, and Be(s)/I(s) can be approximated as
B,~(s)
IS)
-G(s)F(s).
(6.17)
lFe
The filter F(s) therefore will contribute significant distortion near saturation. Figure 6-9 shows the Bode plot of the stabilizing filter. At 1 Hz, we see a phase lag from
the filter of 230.
6.3
Flux Estimation Using an Observer
Another way to provide a real-time estimate of the flux is to design an observer to
model the actuator. If we can estimate the current using a model of (4.1), we can
compare this estimated current, 1, with the actual measured current, I. We can then
design a feedback controller to drive the error between the estimated and measured
currents to zero. This controller thereby forces the model to estimate the H and B
218
Bode diagram of F(s)=1/(rs+1)
100
10
0t
10~0
System: sys
-
to
Frequency (Hz) 1
Phase (dg) -23
a) -45
(U
c
-90
2
102
101
100
101
Freciuencv (Hz)
Figure 6-9: Bode plot of stabilizing filter F(s).
that will solve (4.1). Figure 6-10 shows the block diagram of such an observer. The
hat notation (^) used in this section and subsequent sections denotes an estimated
variable.
Actuator Model
NI
NZ
+1+
2
Controller
LdB(H)
Hysteresis
g
Figure 6-10: Block diagram of observer for estimating actuator flux density.
The changing gap can be treated as a disturbance to the plant model. The controller ideally will drive the model to the correct H and $ even with this gap disturbance. The major advantage of using an observer to model (4.1) is that we do not
need explicitly to solve for H in terms of I: the controller 'automatically' finds the
correct input to the model that solves (4.1) if the estimation loop is stable.
One potential concern is that because of the hysteresis nonlinearity, (4.1) will
219
not necessarily always have a unique solution. However, because of the 'wiping-out'
property of the Preisach model [61], we expect the observer to converge to the correct
solution. The wiping-out property entails that previous local extrema in the input
to the hysteresis model are removed from memory when they are exceeded by the
current input. This memory is used to compute the current Preisach output. Thus,
as the input continues to increase or decrease and previous extrema are wiped out,
the Preisach model history will be erased and the output will converge to a unique
solution, even if the initial output was inaccurate.
The benefit to this method compared to the previous method of using a stabilizing
filter is that since the controller is separate from the plant, we can design the controller
to have high gain over a large range of frequencies in order to drive b to the correct
value.
In contrast, the stabilizing filter method required that the stabilizing filter
effectively become part of the plant model itself, which resulted in inaccuracies in the
estimated flux density even at very low frequencies.
6.3.1
Controller Design
To design the controller for the observer, we first need to approximate the plant model.
Rather than using the describing function analysis from Section 6.2.2 to approximate
the hysteresis gain, we instead approximate it with the maximum dB/dH as was done
in (6.15). This will bias the controller design toward a conservative one for a better
guarantee of stability. The plant gain, PI, can then be approximated as
dB
P~-+ dH
2g
max/PO
lFe-
(6.18)
For initial investigation, we implemented the Della Torre model for the hysteresis
relationship, using the parameters given in Section 4.1.2 from the simulation model
used in Chapter 4. The value of dB/dHmax with these parameters is 3.98 T - m/A.
This results inI P = 3164 in at a nominal gap of g = 500 pm. Note that the units of
the plant gain are length (in). This can be seen by inspecting (6.18): the parameter
1Fe for instance has units of meters. This is likewise true of the first term: dB/dH
220
has units of T - m/A and po has units that are the reciprocal of this. This leaves the
variable g, which again has units of meters.
We choose as our controller a simple integrator, C(s) = K/s. This will add 900
of phase loss to the loop. Provided that our loop crossover is sufficiently lower than
the sampling frequency, we should have enough phase margin for stability, since the
only additional phase loss comes from the hysteresis.
We set the nominal crossover
f,
K =
to 2 kHz. The gain K is then set to
fc =
IPI
3
.9 7 m-s1.
(6.19)
Our initial controller is therefore
C(s)
6.3.2
K
S
s
.
(6.20)
Simulation Results and Analysis
Using the controller in (6.20), we simulated the observer with a 1 Hz sine wave with
3.2 A amplitude as the current input. Figure 6-11 shows the resulting B-I curve along
with the corresponding b-fH curve. We see that the observer flux density estimate is
poor near saturation and when traversing from the demagnetized state. The reason
for this is because in these regions, dB/dH is very low, which results in a low plant
gain and consequently a low loop gain. Thus, we have low control authority in these
regions and poor following error.
The low control authority also manifests itself when plotting the estimated current.
Figure 6-12 plots the estimated current along with the input current. The controller
is unable consistently to drive the error between the two to zero.
The minimum dB/dH, denoted dB/dHmn, is equal to po. This can be seen from
(4.3) when dM/dH approaches zero. This occurs at saturation for instance, since
all the magnetic dipoles are already aligned with the external magnetic field, and so
no additional magnetization can occur. For the Della Torre model, this also occurs
at the demagnetized state: substituting zero for H in (3.12) and (3.13) results in
221
1.5
E
A
i
v.
Regions of low dB/dH
0.5 F
0
-0.5 F
-1
-1.5
-4
h V. fi
-2
0
current [A]
2
4
E
0.5 F
C
0
a.'
x
~0
-0.5
-1
-6
-4
2
-2
4
6
3.97
Note
magnetic field [A/m]
Figure 6-11: Simulated $-I curve from observer with controller C(s)
errors where dB/dH is small.
4
-
--i
2
(D
0
Q)
-2
-4
0
0.2
0.4
0.6
0.8
1
time [sec]
Figure 6-12: Simulated input current (I) and estimated output current (I) from
observer with controller C(s) = 9
dM/dH = 0. In real materials, however, dM/dH 4 0 at the demagnetized state. This
can be accounted for in the Della Torre model by adding a reversible or anhysteretic
222
component to the magnetization [8, 33]. Reversible magnetization accounts for the
fact that some of the energy transferred by an applied field is stored and can be
completely recovered when the applied field is returned to its original value [33].
There are different ways in which the reversible component can be implemented: it
can be implemented as a single-valued function of the applied field [33], or it can be
implemented as a single-valued function of the magnetization [8, 21, 33], for example.
Revising (6.18), we can write the minimum plant gain, Pmm, as
+lFe= 2g +1Fe.
= dBH 2,,yg
IP|minIpmn=dH
minO
(.21)
The minimum plant gain is 0.101 m at a nominal gap of g
=
(6.21
500 lim. This is a factor
of 30, 000 less than the maximum plant gain! The upshot is that the minimum loop
gain will likewise be a factor of 30,000 less than the maximum loop gain. The cutoff
frequency at the minimum loop gain is estimated to be
fc = KIPin
27r
=
0.0638 Hz.
(6.22)
As this is well below the signal frequency of 1 Hz, our controller will have limited
effectiveness and our observer performance will suffer when dB/dH is low. We will
revisit this topic in Section 6.8.
6.3.3
Controller Design with Higher Gain
We can attempt to increase the controller gain so that we have more control authority
when the plant gain is low. We must be cautious that we do not increase the controller
gain too much lest we compromise stability.
We increased K by a factor of 20, so that our updated controller, C2 (s), is
C2(S)
--
S
S
.
(6.23)
The resulting B-I curve of the observer for the 1 Hz sine wave input is shown in
Figure 6-13.
The observer shows much better performance than with the original
223
controller. However, one can still see that performance is degraded at the demagnetized state and at saturation.
- -
- - -
--
-
- 1.-1 6
1.16
-/ --
1.14
I 1.14
I
1.12
B v.I
10
1.08
-
1.06
4L.
-
1.5
-
I
1.04
1.02
X
0.5
2.5
5
3
-I
0
-
-.
-
-
-
-
-
E
-1
0.14
-1.5
_ _ __ _
4
-2
n01-
_
0
current [A]
2
I
0.05
0
-0.05
'
-0.1
\
I
%I
-0.15
-1
\
---------- -I
-0.5
-
-
0
0.5
-
Figure 6-13: Simulated b-I curve from observer with controller C(s) =9.4
Figure 6-14 shows the input current and estimated observer output current in the
top graph. The estimated output current tracks the input current more accurately
with the higher-gain controller. The bottom graph shows the error between the two.
The error can be seen to be larger at the initial state (65 mA error) and at saturation
(14 mA error). This error will increase further as the input frequency increases and the
control authority deteriorates. Increasing the controller gain much beyond a factor
of 20 of the original gain led to stability problems.
6.4
Observer with Adaptive Controller Gain
To resolve the difficulty of providing an accurate flux estimate in the face of a wideranging plant gain, we designed a controller with an adaptive gain dependent on the
224
Observer input current and output estimated current
5
-
-
-
Coj
..
wa
-5'i
0
0.4
0.2
0.6
0.8
1
0.6
0.8
1
Error
0.1
S
0
-0.1
0
0.4
0.2
time [sec]
Figure 6-14: TOP: Simulated input current (I) and estimated output current (I)
from observer with controller C(s) = 7-4; BOTTOM: Error between I and I.
plant gain.
The Della Torre Preisach model implementation outputs dB/dH directly, which
can
makes it particularly well-suited for implementing a dynamic gain controller. We
to
design our adaptive controller, Ca(s), such that the gain is inversely proportional
dB/dH, to wit,
Ks/d
Kd
Ca(s)==
where K, is a static gain, and
Kd
S
_
K
5
/dH
(6.24)
S
is the dynamic gain equal to K/(dB/dH). With the
gain inversely proportional to dB/dH, we will have high controller gain when dB/dH
is low, giving us more control authority when we need it most, and low controller gain
when dB/dH is high, helping us to maintain stability. The loop gain should remain
approximately constant even while the plant gain is changing.
The loop gain as a function of frequency,
ILTI
Ks/
f,
can be approximated as
dB 2g)
2irf kdH po'
ILTI
K. g
irf po
225
(6.25)
From this, we can solve for K, by setting the loop gain to 1 at
K8 =
For a loop crossover frequency of
6.4.1
fc =
f
=
f,
(6.26)
.fp
2kHz, K, = 15.8 T/(A - s) at g
=
500 pm.
Simulation Results
The Simulink model of the adaptive gain observer is shown in Figure 6-15. The gain
K, gets divided by dB/dH, which is fed back from the Preisach model output. A
delay of one time step is placed into this feedback path to break the algebraic loop.
For our hysteresis model, we used the model fitted to experimental data described
in Section 6.2.1. In (6.26), we replace po/g with 2m/N, with the result that K, = 21.1.
Figure 6-16 shows the observer estimated flux density plotted against input current
for a 1 Hz sine wave with 2.9 A amplitude. Shown in the same plot is the experimental
B-I data from Figure 6-1 that was used to fit the hysteresis model. The dynamic
gain observer output matches the experimental data well.
Also shown is a graph
zoomed in on the B-I initial trajectory from the demagnetized state. The observer
with adaptive controller gain is seen to have superior performance to the observer
with static gain.
Figure 6-17 shows the input current and estimated observer output current in the
top graph. The bottom graph shows the error between the two. The error has been
much reduced: the maximum error is 1.4 mA. The largest errors no longer occur at
the demagnetized state and at saturation, showing that the adaptive gain controller
resolves this problem effectively.
6.4.2
Preliminary Experimental Results
We next implemented the adaptive gain observer in real-time using the measured
current to the prototype reluctance actuator as the input current. We clamped the
actuator target to the stator with 500 pm plastic shims between the target and stator. The actuator was driven with a Kepco BOP 36-12M amplifier [50] the current
226
E
E
x +
E
z zz
4'
Figure 6-15: Simulink model of adaptive gain observer.
227
B v. I
1.5
experimental
simulation
-
0.5
E
0
-0.5
0.1
-1
0.05
\
-1.5
3
-2
-1
-
0.15-
-E--
------
----
0-
% 1
current [A]
-0.05
%
%
0.1
0.15/
%4
%I
-0.2
0.2
0
02
0.4
Figure 6-16: Simulated adaptive gain observer estimate of flux density versus input
current.
Observer input current and output estimated current
51
C
0
-5
2
-2
0
0.5
1
1.5
2
0.5
1
time [sec]
1.5
2
Xl10-
0
Figure 6-17: TOP: Simulated input current (I) and estimated output current (I)
from observer with adaptive gain controller; BOTTOM: Error between I and i.
was measured with a sense resistor. A picture of the clamped actuator is shown in
Figure 6-18.
Figure 6-19 shows the real-time estimated flux density, B, of the observer for
228
Figure 6-18: Clamped actuator with sense resistor shown.
a series of measured input current sine waves of varying amplitudes each driven at
1 Hz. The graph on the right shows the corresponding estimated B-H curves from
the observer.
If we take one of the estimated B-H curves from Figure 6-19 and inspect it more
closely, we see that there are 'whiskers' near the major reversal points, as shown in
Figure 6-20.
These whiskers are the result of higher frequency variations in the measured current, e.g. from noise sources. These variations cause H to reverse direction, resulting
in the traversal of tiny minor loops, giving the appearance of whiskers. Figure 6-21
shows the measured current that gave rise to the whiskers in Figure 6-20, filtered at
200 Hz. The inset graph shows the current zoomed in near the maximum current,
showing that the higher-frequency variations cause the current to change direction.
229
BV. Im
2
1.5
E
1.5
1
E
1
0.5
0.5
0
0
1 -0.5
Ii -0.5
E
=
E
Z5
a
-1.5
-1
-1.5
-9
-
-2
-4
E3V.
BvH k
2
-2
0
2
measured current [A]
4
-1500
-500
0
500 1000
estimated magnetic field [A/m]
-1000
1500
Figure 6-19: LEFT: Real-time observer estimates of flux density versus measured
current; RIGHT: Real-time observer B-H curves.
bv. f
0.5
E
'whiskers'
a
0
E
C',
a)
-150
-100
-50
0
50
100
estimated magnetic field [A/m]
150
Figure 6-20: Real-time observer B-H loop showing 'whiskers' near reversal points.
This explains why the minor loops only occur near the major reversal points of the
B-H curve: since dI/dt of the fundamental frequency component is close to zero near
the major reversal points, the higher-frequency variations cause tiny reversals in the
direction of the current, and hence likewise in H. In contrast, during the high dI/dt
portions of the fundamental-frequency sine wave, the higher-frequency variations in
current are not large enough to reverse the direction of the current.
230
Measured actuator current
1
9
-
=%
0.5
96-.0.96I
-
------
-
C
U)
0.94
-0.5
0.9
-
0.92
0.88
0.86
-
1
0
Figure 6-21
0.2
0.6
0.4
time [sec]
0.8
s...
0.84 .
0.8
0.85
0.9
Measured actuator current filtered at 200 Hz showing high-frequency
reversals.
6.4.3
Incorporating a Gap Disturbance
We next wanted to evaluate how our adaptive-gain observer handled gap disturbances.
For a real-time hysteresis model to be effective for a motion control system, it must
be able to estimate flux density accurately in the face of gap disturbances.
With our clamped actuator setup, we input a simulated gap disturbance to the
observer model. The simulated gap disturbance is the same that was used for the
simulations in Section 4.3 and that is shown in Figure 4-13. The observer was run in
real-time and used the measured actuator current as its input.
Figure 6-22 shows the real-time estimated flux density of the observer for a series
of measured input current sine waves of varying amplitudes each driven at 1 Hz and
with the simulated gap disturbance input to the observer. The graph on the right
shows the corresponding B-H curves from the observer.
In Figure 6-23 we plot the outermost B-Im loop of Figure 6-22 zoomed in near the
maximum flux density so that the effect of the gap disturbance can be more easily
discerned. The gap disturbance effect is greater in this region because the flux density
deviation from a gap disturbance increases with nominal flux density level (see (4.34)).
Moreover, since dI/dt is lower near the maximum current, the flux density deviations
from gap disturbances will appear more pronounced. In other words,
not overwhelmed by OB/IAI in this region.
231
9B/&gAg is
$ V. I1
H
1
1
E
E
0.5 F
0.5 FU,
2
78
$1 v.
1.5
Xi
0
0
Ca
.9
V5
x1
-0.51-
-0.51F
-1
-1
-1.5 '-4
-2
0
2
measured current [A]
-1.5'
-300
4
-200
-100
0
200
100
estimated magnetic field [A/m]
300
Figure 6-22: LEFT: Real-time observer estimate of flux density versus measured
current with simulated gap disturbance; RIGHT: Real-time observer B-H curves
with simulated gap disturbance.
B
V. Im
1.4
C
-a
X
1
E 0.8
U)
0.6
2
2.5
3
measured current [A]
3.5
Figure 6-23: The effect of a gap disturbance on the observer estimated flux density.
6.4.4
Numeric Problems with the Della Torre Model
While the initial results for handling a gap disturbance looked promising, the observer
had difficulty with larger gap disturbances. Figure 6-24 shows two B-H curves with
identical current inputs: the blue curve is the B-H curve with the original gap disturbance; the green curve is the B-H curve with twice the original gap disturbance.
Since the current inputs were the same for both curves, we expect to trace the same
232
primary B-H loop in both cases. Instead, however, we see some discrepancy between
the two. The whiskers on the B-H curve with twice the gap disturbance are also
much more pronounced.
1.5 -
h v. I
original gap disturbance
- 2 x gap disturbance
C
x
0.5
~0~0
c -0.5
E
-1
-1.5
-150
100
50
0
-50
-100
estimated magnetic field [A/m]
150
Figure 6-24: Comparison between observer B-H loops for different gap disturbances.
We suspected the culprit to be numeric problems with solving the Della Torre
hysteresis model. To test this hypothesis, we ran simulations of just the hysteresis
model without the observer using the H data from the green data set in Figure 6-24
as the input. We ran this simulation multiple times, using different numeric solvers.
Figure 6-25 shows the resulting B-H curves for the solvers 'odel', 'ode2', and 'ode3'.
We see that the B-H loops depart significantly from one another, indicating that
numeric problems are playing a role. We also found that changing the time step
affected the hysteresis model output.
We posited that the reason the numeric solver had trouble accurately solving the
Della Torre model is because the Della Torre model outputs dB/dH rather than
outputting B directly. When the observer sees a relatively large gap change, this
results in a large change in the value of H as input to the hysteresis model. Since
dB/dH has a highly nonlinear dependence on H, this can result in an inaccurate path
taken from the original value of B to the new value of B after the gap change. To
demonstrate this, we ran a simulation with the model described in Chapter 4 where
we ramped the input current to a constant level and then applied a -200 pim step to
233
B v.
0.8
0.6
H
odel
-- ode2
- -ode3.......
EO.4x
-0.2
-150
-
0.2 - -
-100
-50
0
50
magnetic field [A/m]
100
150
Figure 6-25: Comparison of different numeric solvers for Della Torre hysteresis model
for same H input.
the gap. We used the Chapter 4 model instead of the observer model from this section
so as to isolate actuator model effects from controller effects. Figure 6-26 shows the
current profile and gap profile for the simulation.
Simulated gap profile
Simulated current profile
1.6
550
1.4 -00
500
1.2
450
E
0.8
400
CL
0a
350
.
0.4
300
0.250
0
0.2
0.4
0.6
time [sec]
0.8
0
1
0.2
0.4
0.6
time [sec]
0.8
1
Figure 6-26: Simulated input current and gap step to actuator model.
Figure 6-27 shows the resulting B-H curve.
The gap step results in a sharp
increase in the value of H. This results in a sharp reduction in dB/dH, since at
the new value of H the Della Torre model is in the saturation region where dB/dH
is small. The integration routine for calculating B from dB/dH is unaware of the
nonlinear nature of dB/dH and assumes that dB/dH is constant between the previous
234
value of H and the current value of H. This results in a large error in the resulting
B. Note that while the Della Torre model thinks it is in the saturation region, the
resulting B is about 0.7T, which is far too low for the saturation region. We can
see then how gap changes that result in large changes in the value of H can lead to
errors in the estimate of B. Sudden large changes in current can produce the same
phenomenon.
B v. H
0.8
Gap change occurs here
0.6
CO0.4
-
-
.
.
-
0.7
0.
0
20
40
80
60
100
120
140
magnetic field [A/m]
Figure 6-27: Simulated B-H curve showing the effect of a -200 im on the Della Torre
hysteresis model.
One way to mitigate this problem is by reducing the controller gain of the observer.
From the block diagram in Figure 6-10, we can write for a sudden change in gap
AH = Ce = CN [I - (I + AI)] ,
(6.27)
where i is the observer estimate of the current before the gap change occurs, Al is
the change in I resulting from the sudden gap change, and e is the error between
the input current and observer estimated current. By reducing the gain of C, the
error arising from a sudden gap change will translate into a smaller AH as input to
the Della Torre model. A smaller AH will lead to a smaller error in the integration
routine.
Simulations showed that reducing the controller gain was indeed effective in handling larger gap disturbances. However, a more robust solution is preferred.
235
6.4.5
Hui Hysteresis Model Implementation
The Hui implementation of the Preisach model as described in Section 3.3.2 has the
advantage over the Della Torre implementation of having as its output B(H) rather
than dB/dH(H). Using the Hui implementation should therefore avoid the numerical
problems that come with integrating a nonlinear dB/dH(H) relationship.
We implemented the Hui model into our adaptive observer.
To do so, we im-
plemented lookup tables for the ascending branch and the descending branch of the
major B-H hysteresis loop. One downside of the Hui model compared to the Della
Torre model is that we no longer get dB/dH for free, which we need for our adaptive
controller gain. We compute dB/dH by differentiating (3.29-3.31) with respect to
H. We also compute the derivatives of the functions T (3.28) and F+ (3.25) and F(3.27) with respect to H. This allows us to express dB/dH in terms of dB'/dH and
dBd/dH, the derivatives of the ascending and descending branches of the major loop
with respect to H, respectively. We implement dBa/dH and dBd/dH as additional
lookup tables.
We simulated the observer response with the Hui implementation of the Preisach
model for a 1 Hz sine wave input current of 2.9 A amplitude and twice the original
gap disturbance. Figure 6-28 shows the resulting B-I and B-H curves. These results
look much more promising than those with the Della Torre model for the same gap
disturbance. Compare the B-H plot here with that shown in Figure 6-24 for instance
(although note that the current inputs are different for the two cases).
We simulated with larger gap disturbances as well. Figure 6-29 shows the simulated B-I and B-H curves for a gap disturbance five times the original (this gives a
peak gap disturbance of 41 pm, about 8% of the nominal operating gap). The plots
look reasonable, although we note the spikes present in the B-I curve.
These are
artifacts of the observer controller and are not genuine responses of the magnetic
flux to the gap disturbances. When we are near the high-slope region of the B-H
curve (H is near the coercive field), even a relatively small change in H from a gap
disturbance can result in a large initial change in B. These large sudden changes
236
i3v. f
fV. I
1.5
1.5
1
1
E
E
- 0.5
:L-
x
2
-
0.5
x
0
a)
Ii
E -0.5
E -0.5
1 --
-1 --1.5
-3
0
4
Ca
'
-2
'
-1
'
0
current [A]
'
1
'-1.5
2
-1000
3
500
0
-500
estimated magnetic field [A/m]
1000
Figure 6-28: LEFT: Simulated Hui-model observer estimate of flux density versus
input current with twice original gap disturbance; RIGHT: Simulated Hui-model
observer B-H curve with twice original gap disturbance.
in B result in the spikes seen in the B-I curve. The controller eventually recovers
and settles to the correct B-H operating point. However, these spikes will result in
errors in the estimated flux density. We discuss this phenomenon in more detail in
Section 6.8. One possible solution is either to filter B or to filter the gap measurement
itself. Since the hysteresis model only needs to be accurate for low frequencies (the
sense coil will measure the flux for higher frequencies), this may be an acceptable
solution. Another possible solution is somehow to feedforward the gap measurement
so that the observer can anticipate any sudden changes and react accordingly.
We found in simulation that if the gap disturbance becomes too large, the observer
risks going unstable.
Nevertheless, the Hui implementation performs significantly
better than the Della Torre model in this regard, which had stability problems even
at only twice the original gap disturbance. Lowering the controller gain can help avoid
the problem of instability. While the Hui model implementation improved things, a
more robust solution of dealing with gap disturbances is still desired.
237
$3V.
1.5
I
$3V.
1.5
E
A
f
E
0.5
0.5
CC
X
X
0
0CaCa
E -0.5
-
E -0.5
Artifact of controller
-1 --
-1.5
-3
1
'
-2
-1
0
1
current [A]
2
-1.5
-1000
3
-500
0
500
estimated magnetic field [A/m]
1000
Figure 6-29: LEFT: Simulated Hui-model observer estimate of flux density versus
input current with five times the original gap disturbance; RIGHT: Simulated Huimodel observer B-H curve with five times the original gap disturbance.
6.5
Flux Estimation Using Interpolation
One way to avoid stability problems arising from the gap disturbance is to interpolate
among different models at different nominal gaps rather than to input the gap directly
as a disturbance to the model. Figure 6-30 shows the Simulink block diagram of an
interpolation scheme we used for simulation. There are three observers, each with its
own hysteresis model corresponding to a different nominal air gap. In this simulation
model, the gaps for each model are 450 im, 500 pm, and 550 pm, respectively. The
simulated actuator current is the input to each of the three observers, so each observer
estimates the appropriate flux density for that input current at its respective operating
gap. The final estimate of flux density is then obtained by taking the simulated gap
measurement and interpolating among the three observer models. We use a linear
interpolation routine.
Figure 6-31 shows the flux density estimated by the interpolation scheme plotted
against the input current and with the original gap disturbance. The input current
was a 1 Hz sine wave with 2.9 A amplitude. The nominal gap was 500 Pm. We also
show the flux density estimated by the adaptive-gain observer using the Della Torre
model from Section 6.4 for the sake of comparison.
238
The two methods show good
NI
B
Flux Observe7450 umn
Three observers: three hysteresis
models for three different gaps
currenit
--
Ii']
-
t'
NI
N--
current input
B
I
N
Flux O15ever
un
Gap interpolation
NI
x
B
Lxd at
Flux 0berver
550 un
ydat
y
Best
estimated B
Tae
gap
Dynamic
Gap measurement
[Time LS-erry]
1
gap distence _
gain
I
_C_
450 um
_C_
500 um
g
[550 urn
Figure 6-30: Interpolation scheme for estimating actuator flux density.
agreement.
Figure 6-32 shows the deviation in estimated flux density, AB, between the two
methods. The maximum AR is 1.5 mT.
Figure 6-33 shows the comparison of the two methods when the gap disturbance
is twice the original gap disturbance and when the gap disturbance is five times the
original gap disturbance.
For the simulation results from the direct-gap observer
method, the sampling rate was increased to 100 kHz and the controller gain was
reduced to maintain stability.
The B-I curves are zoomed in near saturation to
emphasize the effect of the gap disturbance and to highlight the discrepancies between
the two methods. The bottom plots show the A$ between the two methods. For the
simulations with twice the original gap disturbance, the maximum AR between the
two methods is 4.6 mT and for simulations with five times the original gap disturbance,
the maximum AR between the two methods is 19.6 mT.
239
-
- -
-
- -
-
$ v. I
1.5
interpolation method
-direct gap method
I
E
0.5
C
0-14-
4
-.
E
--- - - - - - - -
-
x
-0.5-
t
1
2I
-1.5.2
-3
-2
-1
0
current [A]
1
10
U
0.6
2
1.5
V
2.5
3
1
Figure 6-31: Simulation with gap disturbance: comparison of b-I curves using gap
interpolation scheme (blue) and adaptive-gain observer with gap disturbance applied
directly (green).
2x
10-3
1.5
E 0.5
- ----
- 0 .5
- -
- -
-
-
0
<l
-1
0
02
0.4
0.6
time [sec]
0.8
1
Figure 6-32: Ab between interpolation model simulation and direct gap model simulation both with original gap disturbance.
The foregoing simulation results suggest that interpolation may be a reasonable
way to deal with the gap disturbance while avoiding instability. The primary downside is that it requires additional models operating in parallel, which reduces the
computational efficiency compared to the direct gap method. If we require a larger
operating range or additional accuracy, models for additional operating gaps must be
added, further reducing the computational efficiency.
240
fv. Ifor 5X
$ v. Ifor2 Xg9
--
method
-interpolation
- - - direct gap method
interpolation method
--- direct gap method
0.5
0.5'
2
1.5
1
2.5
1.5
1
3
2
2.5
3
I [A]
I [A]
5
9d
1.5
1.5
0.02
x 10-3
0.01
-
<0-
a
0
-0.01
-5
'
0
0.2
'
0.6
0.4
time [sec]
0.8
-0.02'
0
1
0.2
0.6
0.4
time [sec]
0.8
1
Figure 6-33: Comparison between interpolation method and direct gap method. (Simulation). LEFT: twice the original gap disturbance; RIGHT: five times the original
gap disturbance.
Another disadvantage of interpolation is that it does not capture the hysteretic
behavior that results from a changing gap.
Since a gap disturbance will change
the material (H, B) operating point needed to solve Ampere's Law, the relationship
between B and g is ultimately hysteretic.
However, the close match between the
interpolation simulations and the direct gap method simulations suggest that this
hysteretic relationship may be negligible, at least for the scale of gap disturbances
with which we are concerned and for the material hysteresis we are modeling.
We recommend that if the interpolation method is selected to handle gap disturbances, the hysteresis relationship between B and I be modeled directly for each
operating gap rather than modeling the hysteresis relationship between B and H and
using an observer to obtain the B-I relationship at each gap. The observer structure
becomes superfluous in such a case, since with no gap disturbance there is a one-toone relationship between the B-I curve and the B-H curve. Moreover, since the B-I
curves can be measured directly unlike the B-H curves, the potential for accurate
calibration and accurate model-fitting is much greater.
241
6.6
Observer with Sheared Hysteresis Model
In this section, we present a novel way for designing an observer structure that can
handle the gap disturbance while avoiding the stability problems discussed in Section 6.4.
Instead of modeling the material B-H relationship directly, we model a
sheared version of the B-H relationship. The sheared model does not have the sharp
changes in dB/dH that the material B-H relationship has. This makes the observer's
estimation problem much easier.
We begin with a change of variables in the model given by (4.1). We define the
variables
92
(6.28)
= 9-goff,
and
H2 =
H +
,(H)
(6.29)
A01Fe
where gff is some constant offset. It can be chosen to be go for example. Figure 6-34
shows how g 2 is related to g and gff, where goff is arbitrarily chosen for the sake of
g
Figure 6-34: The relationship between 92 , g and
actuator.
242
g
)
illustration.
goff is shown on the reluctance
The variable H2 is offset from H by the quantity 2gffB(H)/( 1aolFe). The relationship between H and H2 is demonstrated in Figure 6-35. For example, if we
take a particular test point Hp, Bp from the B(H) curve, we obtain H2 , by adding
2gOffB/(polFe) to Hp. The green curve is the B(H2 ) major loop obtained in this way.
The parameter goff will determine the slope of the B(H2 ) curve.
B
B(H) curve
B(H2) curve
H ,B
/P
H/,B
-I P
H
H
H
2p
P
Figure 6-35: The variable H2 is obtained by adding
If we substitute the definitions of
lFe
H2 -
1FeH2 -
2nB)
I1o1Fe
92
29-ff
to H.
and H2 into (4.1), we can write
+ 2(92+
Ao
gof) B
2goffB+ 2gB + 2goffB
Ao
/to
go
1FeH2
+22B
Po
=NI,
=NI,
=NI.
(6.30)
in
Equation (6.30) is equivalent to (4.1) except that instead of the being expressed
terms of the variables g and H, we have expressed it in terms of the variables 92
243
)
and H2 . The problem of identification is now to identify the hysteresis model B(H2
rather than B(H).
Suppose we choose gff = go. Then at the nominal gap, g2= 0, and we can write
(6.30) as
(6.31)
lFeH2= NI,
or
NI
(6.32)
H2 = --.
'Fe
At g = go then, H2 is proportional to the current, and we can identify the B-H2
relationship simply by identifying the B-I relationship at go. This is one advantage
of this method: we are able to identify the hysteresis model directly from the measured
B-I data rather than having to estimate the material B-H relationship.
As we demonstrated in Section 2.4.4, the B-Ha relationship is a sheared version
of the material B-H relationship, where Ha is the applied magnetic field and is equal
to NI/lFe. We note from (6.31) that this is identical to the expression for H2 at
g = go. The B-H2 relationship is thus a sheared version of the B-H relationship.
Consequently, the hysteresis model of B-H2 will be much more linear than a model
of B-H and will have less drastic changes in slope.
6.6.1
Controller Design
Figure 6-36 shows a block diagram of the observer structure with the sheared hysteresis model (SHM), denoted as B(H2 ). The variables i and b are the estimated current
and estimated flux density from the observer, respectively. Likewise, the variable H 2
is the value of H2 estimated by the observer.
Our plant now takes as its input the variable H2 rather than H. Our plant gain
is
dB 292
+ lFe|P = d
d H2
(6-33)
po
If we design our controller around the nominal operating gap, then
92
= 0 and
|PI = 1Fe. If we keep our controller as an integrator, we can use (6.19) to solve for
244
Actuator Model
FFe
NI
|H2+
+
NI
Controller
a
oBn2g
g
Sheared
Hysteresis Model t
t
Be
90
Figure 6-36: Observer block diagram with SHM.
the integrator gain, K, as
(6.34)
K = 27r,
lFe
where
f, is a desired crossover frequency. For f, = 2 kHz, K = 1. 26 x
105 M-Is-1,
and our controller is
C~)=K =1.26 x 105
s
s
6.6.2
(.5
Simulation Results
For simulation, we fitted the Hui Preisach model to the B-I curve generated by the
adaptive-gain observer in Section 6.4 and represented by the green curve in Figure 616. We used the simulated curve for fitting rather than the experimental data so that
we could attempt to reproduce the simulation results of the adaptive-gain observer
with the SHM observer and compare the two methods directly. Figure 6-37 shows
the simulated flux density estimated by the SHM observer plotted against the input
current with the original gap disturbance present. For comparison, the simulated B-I
curve from the adaptive-gain observer is also shown. The match between the two is
245
very close. The maximum deviation is less than 3 mT.
-
-
V. I
1.5
sheared model observer
- - - unsheared model observer
E
X
-0.50*-
-3
-2
S I
-
1.2
12
-1.58
1
.
cu rrent [A]
- - - - - -
-
EI
II
U
0.6
1
1.5
2
2.5
3
Figure 6-37: Simulation with gap disturbance: comparison of B-I curves using
sheared-model observer (blue) and adaptive-gain observer (green).
Figure 6-38 shows the simulated B-I curves generated by the SHM observer for
the original gap disturbance and for gap disturbances of two times, five times, and
ten times the original gap disturbance.
We note that the SHM observer did not
run into any stability problems in generating these curves: we did not have to lower
the controller gain or increase the sampling rate to maintain stability. We also note
that there are no sharp spikes evident in the B-I curves as there were for the Hui
implementation of the adaptive-gain observer shown in Figure 6-29.
Experimental
results with our testbed for the SHM observer will be presented in Chapter 10.
6.6.3
Incorporating Eddy Currents and Feedforward Control
We can use the observer to model the effect of eddy currents as well. We use (5.28) as
the new actuator model we want to estimate. For Hd1 we use (5.13), and for Hc
we
use (5.38). As both of these equations take dB/dt as their input, we differentiate the
hysteresis model output. Figure 6-39 shows a Simulink block diagram that includes
eddy current effects. There are now two new paths from the hysteresis model output
to the summing junction of the NI estimate representing the classical and excess eddy
246
v.
$
I with original
fv. I for 2 x gd
9d
-
1
0.5
00
-0.5
-2-3
-2
1
0
I [A]
B v. I for 5 x 9$
-1
2
3
3
2
I A]
v. I for 10 x gd
0.5-
0.5
0
0
-0.5
-0.5 -
-11
33
-2-1012 3
1.-3
2
0
-1
I
1
2
.
3
I [A]
[A]
Figure 6-38: Estimated flux density versus input current simulated for SHM observer
with different gap disturbances (lx, 2x, 5x, and 10 x original gap disturbance).
currents. The classical eddy current path is shown in blue. The excess eddy current
path is shown in red. The classical eddy current path is straightforward: dB/dt is
simply multiplied by the constant oed2 lFe/12. For the excess eddy current path, we
use an embedded Matlab function routine to solve the quintic equation of (5.38).
We also add a feedforward path to the input of the hysteresis model for improved
performance. This path is shown in green in the figure. At the nominal gap, H2 is
equal to NI/lFe, so we can simply feedforward the input signal I multiplied by N/iFeThis is another advantage to using the SHM over the unsheared hysteresis model: it is
more difficult to add an effective feedforward path to the unsheared-model observers
since the relationship between the observer input I and the hysteresis model input H
is highly nonlinear. A feedforward path with a simple gain is in that case insufficient.
Simulation results for the SHM observer with eddy current effects included were
already presented in Section 5.4.1, so they are omitted here.
247
6.7
Flux Estimation Using the Chua Model
The Chua model was not investigated in depth for the purpose of real-time estimation
owing to its inability to track minor loops accurately (see discussion in Chapter 3).
For the sake of completeness however, we include some brief remarks here.
6.7.1
Hysteresis Model Identification
We fitted the Chua model to experimental B-I data taken from the reluctance actuator driven with a 10 Hz sine wave with 2.9 A amplitude. We defined the functions
f
and g needed to identify the Chua model in Section 3.4. Here we rename g as G to
avoid confusion with the gap variable. Figure 6-40 shows the
f-
and G
1
functions
we extracted from the data. The w function, also defined in Section 3.4, is set to
dI/dtl /(27rfolo) for frequency-independence, where fo = 10Hz and 1o = 2.9 A.
248
Ife
Hexc
Excess eddy
current path
'f
""n
E mbedded
MATLAB Function
Feedforward path
00
-K-
Derivative
SAfe
sigma E*d^2/1 2*lfe
Classical eddy
H2
cre path
(D
._N
'-1
KH
Current
Input
N
K
t
ss
integrator
CD
0:
CD
n
gap disturbance
d
Product
2/muO
+K
1.5
0.06
1
0.04-
0.5
0.02
0
0
-0.5
-0.02
-1
13
-
-0.04
-1
I
2
[A]
Figure 6-40:
6.7.2
-2%0
3
-50
dBldt [Tisec]
f-
1
and G-
1
50
100
functions for Chua model.
Experimental Results
Figure 6-41 shows the flux density estimated by the Chua model when driven with
a sine wave of the same amplitude and frequency as that used to identify the Chua
model. The Chua model output shows reasonable agreement with the experimental
data.
However, the match suffers somewhat near saturation.
The reason for this
is because the nonlinear relationship between I and B results in the output signal
having a different frequency content from the input signal. This results in dB/dt not
being perfectly proportional to dI/dt in the measured data. However, the choice of
the w function in the Chua model forces the model dB/dt output to be proportional
to dI/dt. As a result, the Chua model output deviates from the measured data.
Figure 6-42 shows a minor loop from the Chua model compared to experimental
B-I data. We can see that the match is significantly worse than for the major loop.
From the discussion in Section 3.4, this was expected.
For the sake of comparison, we investigated the Chua model performance when
the w function was chosen to be equal to 1. This results in a model that is frequencydependent: it will exhibit loop-widening with increasing frequency. Figure 6-43 shows
the Chua model output for the major loop compared to the measured data.
The
actuator was driven at 10 Hz. By comparing this figure to Figure 6-41, we see that
we now get a much better fit between the experimental data and the Chua model
250
1.4I
S
1.3I
1.2
B v.1
1.5
1
-experimental
Chua model output
C
0
-
-0.5
2.8
2.6
I
3
-
0.5
2.4
2.2
% 1
E
0.2
-1
-1.51
-4
-2
0
%
-
0.15
0.1
2
current [A] %
0.05
%0I
-0.05
-
1
-
-0.1
-0.15
-1
-0.5
0
0.5
I
Figure 6-41: Frequency-independent Chua model B-I major loop output compared
to measured B-I data.
output near saturation.
The frequency-dependent Chua model is also better at matching the minor loops
when driven at the same frequency that was used to identify the model. Figure 6-44
shows a minor loop output of the frequency-dependent Chua model compared to the
experimental data. The input signal was 10 Hz. The fit near the minimum flux density
is very good. The fit near the maximum flux density shows some discrepancy, but
the experimental loop and Chua-estimated loop still exhibit the same general shape,
unlike with the frequency-independent Chua model that was shown in Figure 6-42.
The reason the minor loops show a better fit when w is set to 1 is because the Chua
model cannot distinguish between a change in frequency and a change in amplitude.
Both result in decreased dI/dt. Therefore, for the frequency-dependent Chua model,
an input signal of the same frequency but lower amplitude than the signal used to
identify the Chua model will exhibit 'loop narrowing'. However, because minor loops
251
Il
0.54
ii
I
I
I
B v.I
I
I
I
I
I
I
I
0.5
0.48
0.46
-
0.6
0.52
0.4
-
0.44
experimental
Chua model output
0.42
0.2
0.5
1.5
E
0
-0.2
-0.4
-0.6
-0.8
-2
-1
0
1
2
current [A]
Figure 6-42: Frequency-independent Chua model B-I minor loop output compared
to measured B-I data.
are typically thinner than the major loop, this is a case where the loop narrowing of
the frequency-dependent model results in a better fit.
The problem with setting w = 1 is that our model output is now frequency
dependent and will exhibit loop widening or loop narrowing as the frequency increases
or decreases, respectively. Choosing w = 1 only makes sense in applications where
the frequency content is fixed or if we desire the frequency dependence.
6.7.3
Gap Incorporation
One advantage that the Chua model offers is that if the w function described in
Section 3.4 is equal to 1, then the Chua model is capable of incorporating the gap
directly.
This obviates the need for an observer.
To see how, consider again the
differential equation from (3.32), repeated here as
dB(t) = G [H(t) - f(B(t))]
252
(6.36)
'I
ii
I
I
I
I
I
B v. I
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1.3
I
I
I
I
I
1. 25
1.2
-,1.
-
-
1.5
-
-
-
-
experimental
-Chua model output
15
2.2
3
2.8
2.6
2.4
E
Ae
C
(U
--
0
---
"a
%
-0.5
----
-- - - - - -1.5
0
-4
-2
-
0.05
-1
-
-
-------------
2
current [A]
0
'4
I
%
~
-0.4
-0.2
0
0.2
0.4
I
Figure 6-43: Frequency-dependent Chua model B-I major loop output compared to
measured B-I data.
where we have set the functions w and h both equal to 1. We have replaced the
variables u and v with H and B, respectively. If we solve (4.1) for H, we can substitute
the result into (6.36) as
dB(t)
dt
2g
N
G
Fe B
I
- (B(t)).
(6.37)
We now have a state-space equation for B where we can input I and g directly. The
caveat that w = 1 is a significant restriction, however: our model will exhibit loop
widening whether we want it to or not.
253
ii
0.5
I
I
I
I
F
I
0.45
0.6
0.8-
0.6
0.8
1-
I
B v.I
0.6
--
0.4 -- -
0.2
E
ex perimental
-C ua model output
0.35
F
1.4
1.2
1
1
1
0
C
x
0.4
I
-0.2
-0.4
- - - -- - - - - - - - - - I
%
-0.8
-2
-
-0.6
0
-1
1
I
urrent [A]
-0.35
-0.4
-0.45
4%
I
I
4%
-0.5
4%
-1.4
-1.2
-1
-0.8
-0.6
Figure 6-44: Frequency-dependent Chua model B-I minor loop output compared to
measured B-I data.
6.7.4
Chua Model Inversion
We can also show that (6.37) is invertible and can be solved for I. We apply G-
1
to
both sides
G-1 G
-
[N
2g
NI BP0lFe
lFe
f(B(t))
l
,
C-1 (dB(t))
d
G
or
=
I
)
-i (dB(t)
lFe
254
2g B -f(B(t)).
[0lFe
(6.38)
Solving for I gives
I(t) = 2g B + lFe f (B(t)) + G_1 dB),
k dt
N
poN
where G-
(6.39)
is given by (3.35). Equation (6.37) allows for a straightforward implemen-
tation of feedforward inverse hysteresis control: if the desired flux profile is known a
priori, we can use this as the input to (6.37) to determine the appropriate current to
the actuator. Again, note that this result is restricted to the case where w = 1.
6.8
Error Analysis
In this section, we investigate why the SHM observer performs better than both the
adaptive-gain observer from Section 6.4 and the original observer without adaptive
gain from Section 6.3. We show that in the linear case, both the SHM observer and
the adaptive-gain observer respond identically to a gap disturbance. This is done to
demonstrate that the difference between these two observer structures is not the result
of unintentionally designing a better controller for the SHM observer. We then show
conceptually how the hysteresis nonlinearity can easily lead to instability from large
gap disturbances for the adaptive-gain observer. We next analyze the original observer
without adaptive gain from Section 6.3 and show why it leads to poor performance.
Finally, we analyze the SHM observer and examine why the SHM observer performs
better than both the adaptive-gain observer and the original observer.
6.8.1
Linear Analysis of Observer Response to Gap Disturbance
Assume that we have a step change in gap, which we denote as Ag. If the input
current to the observer is constant and the error between I and j before the gap step
change was zero, then the error, e, immediately after the gap step change is
e =-
2BAg
.
A0
255
(6.40)
This can be understood by reference to the block diagrams of Figure 6-15 and Figure 636. We want to determine how this error propagates to the output of the hysteresis
model. In the case of the adaptive-gain observer with the original B-H hysteresis
model, we can write the change in H, AH, as
AH= Kdedt=-
"s/
dB/dH j
B
I
/0
(6.41)
dt,
where we have introduced the controller from (6.24) in its time-domain form. We next
carry out the integration in (6.41) over one time step. With reference to Figure 6-45,
we see that the integration of the error over one time step is 2BAg/PO - T,/2 if we
assume a trapezoidal integration, where T, is the time step. After also substituting
the expression for K, from (6.26), we write AH as
AH
rfBTAg
go
(6.42)
(dB/dH)
j
e
2BAg
Po
to
to+T,
t
Figure 6-45: Error integration over one time step.
For the SHM observer, we can follow the same procedure to derive AH 2 from a
step in gap. For this case, we substitute K from (6.34) to give
A
2
2fBT8 Ag
I1O'Fe
256
(6.43)
We next want to determine the resulting AB in both cases.
dB/dHAH,
Since ABU =
where the subscript u refers to the unsheared model of the adaptive-gain
observer, we can write ABU for the adaptive-gain observer as
(6.44)
irfcBT8 Ag
90
ABU =
For the sheared case, we note that dB/dH 2
=
(olFe)/(2go).
This can be derived
from (6.29) if we assume that H is negligible. Since AB, = dB/dH 2 AH 2 , where the
subscript s denotes the sheared case, we can write AB, for the SHM observer as
AR 8
=
-
27rfcBT8 Ag pO 1Fe
pO'Fe
2go
7rfcBTAg
90
(6.45)
This is the same result as for the unsheared case, showing that in the linear case,
both the sheared and unsheared models respond identically.
6.8.2
Analysis of Adaptive Gain Observer
Because of the nonlinearity, however, the observers can respond very differently in
reality. We can see how there is a risk of instability in the adaptive gain observer by
viewing it from a loop gain perspective. The loop gain of the adaptive gain observer
at time t = to
+ T is
|LT|
-
go !LB
C
s
dB (to +Ts)g)
dH
(6.46)
The reason for the different values of dB/dH in (6.46) is that the value of dB/dH used
for the adaptive gain lags one time step behind the current value of dB/dH. Because
the value of dB/dH(to + T,) can be much larger than dB/dH(to), the loop gain can
be very large at t = to + T,. If the ratio between dB/dH(to + T,) and dB/dH(to) is
large enough, instability can result. Larger gap changes can result in larger changes
in dB/dH from one time step to the next, increasing the risk of instability.
It is perhaps easiest to see this qualitatively. Suppose we are operating at point
a on the B-H hysteresis loop shown in Figure 6-46.
257
The value of dB/dH here is
relatively low. Now suppose we have a gap change in the positive direction. Because
of the low value of dB/dH, the resulting AH for the adaptive gain observer calculated
from (6.42) will be large: since the controller assumes the original low value of dB/dH
for the entire time step, this leads to a large 'overshoot' from point a on the hysteresis
loop to point b in the figure.
The value of B calculated by the hysteresis model
corresponding to the new value of H will be an overestimate. This is the phenomenon
that gives rise to the 'spikes' seen in Figure 6-29. If the gap changes become too large,
the controller is at risk of going unstable. The effect can be reduced by decreasing
the sample time. Another possibility is perhaps to make the adaptive gain somehow
dependent on the gap measurement as well.
Figure 6-46: AH going from low-dB/dH region to high-dB/dH region.
6.8.3
Analysis of Original Observer
What about the original observer without the adaptive gain described in Section 6.3?
We might wonder why the SHM observer performs so much better than this original
model:
after all, the original model did not have a controller gain dependent on
dB/dH, so we should not have to worry about 'overshooting' on the B-H loop.
The reason for the difference is because of how we defined the plant in the two
cases. In our original model, the plant input is H. The plant gain is
dH /-o + l Fe -
|P| =-dB2
258
(6.47)
g
The plant gain is dominated by the dB/dH term. Since this term varies so widely,
we get a large range in plant gain, making good control authority over the entire
range of inputs very difficult. For the hysteresis model identified in Figure 6-2, the
plant gain varies from a minimum of 0.287 m to a maximum of 84.7 m for the range
of inputs simulated, a ratio of nearly 300:1. While the adaptive-gain observer suffers
from stability risks owing to the rapidly changing dB/dH in the controller structure,
the original observer suffers from poor peformance because of the rapidly changing
dB/dH in the plant.
6.8.4
Analysis of SHM Observer
In contrast, with the sheared model the plant gain is much more stable. The plant gain
is given by (6.33). If there is no gap disturbance or gap offset from the nominal, the
plant gain is the constant
1
Fe
Let us consider the case of how the plant gain changes
if there is a gap offset from the nominal. First, we need to determine dB/dH 2 . We
can differentiate (6.29) with respect to B as
2 goff
dH
dH 2
-+=-- lFe
(6.48)
po
dB
dB
The expression for dB/dH 2 is then just the inverse of this, expressed here as
dB
(6.49)
dH2
1
_
= dH+ 2
dB
polFe
.
The term dH/dB is the inverse of dB/dH. For our range of operation, the value of
dB/dH ranges from 3.14 x 10-4 T - m/A to 0.142 T - m/A. The second term in the
denominator is a constant and has a value of 5957 A/ (T
i),
m
where we substituted
N/m for 2goff/po (see Section 6.2.1). Substituting these values into (6.49), we find
that dB/dH 2 ranges from a minimum of 1.09 x 10- T - m/A near saturation to a
maximumm of 1.67 x 10-
4
T - m/A.
Suppose we have a gap offset of 50 pm from the nominal gap. Substituting this
for
92
in (6.33) and using the maximum dB/dH 2 found above, we calculate a plant
259
gain of 0.113 m, a deviation of only 13.3% from the nominal plant gain.
In addition, the SHM observer sees a much more consistent loop gain from one
time step to the next compared to the adaptive gain observer because the controller
is not dependent on a widely varying dB/dH term. Any changes in loop gain come
from changes in the plant gain, which we just showed to be minimal. This stability
of loop gain accounts both for why the SHM observer does not risk instability for
large gap changes in constrast to the adaptive-gain observer and for why the SHM
observer shows good performance over the entire input range in contrast to the original
observer without adaptive gain.
6.9
Summary
In this chapter we developed several methods for modeling the hysteretic behavior
of a reluctance actuator for real-time flux estimation. We designed an observer to
generate an estimate of the flux density without having to solve for the magnetic
field in the steel. However, the flux estimation of the original observer suffered at
saturation and at other regions with low dB/dH because of low control authority
in these regions. To resolve this, we developed an observer with an adaptive gain
controller that linearized the plant over the full B-H region of operation. However,
the adaptive gain observer had numeric stability problems when subjected to dynamic
gap disturbances.
We then designed an observer that used a sheared model of the hysteresis relationship. By taking advantage of the linearizing effect of the actuator operating gap, we
were able to design an observer that can accurately capture hysteretic behavior and is
robust to large gap disturbances. The main advantages offered by the SHM observer
over the other flux estimation methods presented in this chapter are summarized
below:
9 The hysteresis model can be identified directly from the measured B-I data.
e The model exhibits good performance even into the saturation region of the
260
material.
" The model is capable of estimating flux density in the face of large gap disturbances.
" A feedforward path can be easily added for improved performance.
" The SHM observer is less computationally intensive than a flux estimation
scheme that uses interpolation to estimate the effect of a gap disturbance; it
also captures the hysteretic behavior resulting from a gap change, while an
interpolation scheme cannot.
" The SHM observer is numerically stable in the presence of gap changes.
In the next chapter, we present alternative techniques for modeling the reluctance
actuator, including magnetic circuit modeling and Finite Element Modeling (FEM).
The magnetic circuit models can be supplemented with the hysteresis models developed in this thesis to generate an augmented model of the reluctance actuator.
261
262
Chapter 7
Magnetic Circuit Modeling of
Reluctance Actuators
In this chapter, we present an alternative way of modeling a reluctance actuator
using magnetic circuits. Magnetic circuit techniques provide an intuitive method for
modeling and analyzing electromagnetic devices. By analogizing magnetic reluctance
to electrical resistance, magnetomotive force to voltage, and magnetic flux to current,
one can utilize standard electrical circuit techniques (e.g., Kirchoff's Voltage Law
and Kirchoff's Current Law) to solve magnetic circuits. Good references for magnetic
circuit theory and application include [68, 91, 53, 66, 70, 56].
We apply these magnetic circuit techniques to a standard three-pole reluctance
actuator. We first model the three-pole actuator by selecting an appropriate magnetic
circuit network consisting of a magnetomotive force (MMF) source and various reluctances. We then set up the circuit equations via linear network theory. Values for the
reluctances are calculated based on material properties and actuator geometry. Flux
tube methods [74, 59] are used to model fringing field reluctances and leakage field
reluctances. Next we present a method for setting up the circuit equations when the
circuit includes nonlinear elements.
Once the magnetic circuit is specified, we solve the circuit for the air gap flux
density and the corresponding actuator force. This is analogous to solving an electrical
circuit for the current through a particular resistor in the network. We then compare
263
this simulated force to experimental data measured on electromagnetic actuators,
and thus verify the magnetic circuit model and augment it where necessary. We also
present some simulation results using Finite Element Analysis.
7.1
Three-pole Actuator Magnetic Circuit Network
We refer to Figure 2-1 for an illustration of the reluctance actuator stator and target.
By exploiting symmetry between the left and right halves of the actuator, we can
model just one half of the actuator without loss of accuracy.
Figure 7-1 shows a
half-actuator overlayed with a corresponding magnetic circuit.
NI,
Figure 7-1: Magnetic circuit for the half-actuator.
The coil windings are idealized as a lumped MMF source with an MMF value of
NI amp-turns. The reluctances shown in the figure correspond to the following:
0 14: yoke reluctance
'Since the coil windings are equally spaced along the vertical length of the center pole and not
concentrated in one location, a more realistic model of the source would consist of a series of MMF
sources distributed along the center pole instead of a single lumped parameter MMF source. A
single lumped parameter source is used at this point to simplify the analysis.
264
left pole reluctance
*
RpL:
*
2
RpR: right pole reluctance
" Ri: leakage reluctance (cross-slot)
" RgL: left air gap reluctance
* RgR: right air gap reluctance
*
RiL.. .RfLm: m fringing field reluctances from the left pole face to the armature
*
Rf.R1 .RfRn:
n fringing field reluctances from the right pole face to the armature
" Ra: armature reluctance
The magnetic circuit is shown in a standard circuit diagram form in Figure 7-2.
IL
11
~ gfLy 1?ILm
RpL
1?
ZI I
NI
1?pR
]RZ
,R1
RZ]9R2
1ZgR
g
R
nn
Figure 7-2: Magnetic circuit for the half-actuator in standard form.
2
Note that 'right pole' here refers to the right pole of the half actuator in Figure 7-1. With
respect to the full actuator, this corresponds to the left half of the center pole.
265
7.2
Solving the Magnetic Circuit via Linear Network Theory
Figure 7-3 shows the magnetic circuit with the MMF's and fluxes labeled. The node
MMF's (Ti,,), or magnetic potentials, are analogous to node voltages in an electrical
circuit. Note that we arbitrarily set T5 = 0 as the reference node. In a similar manner,
the magnetic fluxes (4)i,), are analogous to currents in an electrical circuit. The force
on the electromagnet target is generated by the gap flux density. Consequently, the
magnetic fluxes of primary interest are
4gL
and
( 1 qR.
PL
'F
ZI
.. ZjLIm
jc0i
+
NI
f'L2...P L
1I
(pR
T
6
XF5
R
]ZRJ
fl?2
..
gR
Figure 7-3: The magnetic circuit with the magnetic potentials
and the magnetic fluxes (Di,,) through each reluctance.
..
IRn
('1y 8 ) at each node
Following the linear network theory procedure outlined in [79] and [66], we can
solve for the flux through each reluctance and for the MMF at each node.
Using
Kirchoff's Voltage Law (KVL) and Kirchoff's Current Law (KCL), we can write the
network equations in general form as
[
P-1
A
AT
0
1
1
L
u
L1
(7.1)
Here, P is a permeance matrix, consisting of the permeances of each reluctance path,
266
where P =
1/Ri.
The matrix A is an incidence matrix that represents the flux
into or out of each node. The vectors 4
and I@ consist of the fluxes through each
reluctance and the MMF at each node (except for node 5), respectively. Finally, 4I,
and T, are vectors corresponding to any MMF or flux sources, respectively.
To understand the relation expressed in (7.1) more clearly, we derive it for the
circuit in Figure 7-3 in the following steps. For the sake of simplicity, we have assumed
only two fringing field reluctances in the following derivation (RfL and lZfR).
1. Write the MMF drop across each reluctance as a function of node MMF's:
Iy =
NI+ Q6 -
TpL
=
1 -
X2,
1
=
'2 -
'5,
fL
=
2 -
TgL
732
XF2 -
XF3)
T4,
XFa
=
3 -
PfR
=
X4 -
XigR
=
T4
X9pR
=
XY5 -
267
1,
XF5
~
'5i
T6.
(7.2)
2. Write (7.2) in matrix form:
NI
1
0
0
0
0
-1
0
-1
1
0
0
0
0
0
0
-1
0
0
1
0
0
0
-1
1
0
0
0
X gL
0
0
-1
1
0
0
0
XF a
0
0
0
-1
1
0
0
0
0
0
0
-1
1
0
0
0
0
0
-1
1
0
0
0
0
0
0
-1
1
Wy
IF,
qjfL
WgR
T pR
L
9'1
92
i
3
(7.3)
5
P6
In matrix notation, we write
Tr = 'Q - Ao'f.
(7.4)
3. By noting that Ps has been selected as the reference (ground) potential, i.e.,
P 5 = 0, we can remove the
5 th
column from A 0 and write
1
0
0
0
-1
-1
1
0
0
0
0
-1
0
0
0
P1
0
0
-1
1
0
0
P
XgL
0
0
-1
1
0
0
'Fa
0
0
0
-1
1
0
P
Tf R
0
0
0
0
-1
0
P6
APgR
0
0
0
0
-1
0
XPpR
0
0
0
0
0
1
Py
NI
qPpL
0
q1
0
PJL
268
2
P3
4
(7.5)
In matrix notation, we write
(7.6)
=rs - A,@.
',
4. Write the constitutive relations between each reluctance MMF and the flux flowing through it. This is analogous to Ohm's Law. Then write these constitutive
relations in matrix form:
D
Pg
0
0
0
0
0
0
0
0
4DpL
0
PpL
0
0
0
0
0
0
0
0
0
P
0
0
0
0
0
0
4
DfL
0
0
0
Pf L
0
0
0
0
0
Xf L
(DgL
0
0
0
0
PgL
0
0
0
0
XIgL
(Da
0
0
0
0
0
Pa
0
0
0
Ta
4f R
0
0
0
0
0
0
PfR
0
0
Xf R
DgR
0
0
0
0
0
0
0
PgR
0
XAgR
4
0
0
0
0
0
0
0
0
PpR
XpR
DpR
I
9pL
(7.7)
In matrix notation, we write
<D = P r.
(7.8)
In general, provided there is no coupling between reluctance elements, P will
always be a diagonal matrix.
5. Combine (7.6) and (7.8) to form the matrix equation <h = P(*L
269
- AL). Then
multiply both sides by P-1 , recalling that Pi
R
1/Rj:
0
0
0
0
0
0
0
0
0
RpL
0
0
0
0
0
0
0
<DpL
0
0
R1
0
0
0
0
0
0
<br
0
0
0
7
R
L
0
0
0
0
0
<bfL
0
0
0
0
RgL
0
0
0
0
(DgL
0
0
0
0
0
Ra
0
0
0
'ga
0
0
0
0
0
0
RfR
0
0
0
0
0
0
0
0
0
Rg9 R
0
0
0
0
0
0
0
0
0
RpR
DpR
NI
1
0
0
0
-1
0
-1
1
0
0
0
0
0
-1
0
0
0
I
0
0
-1
1
0
0
P2
0
0
-1
1
0
0
P3
0
0
0
-1
1
0
P
0
0
0
0
-1
0
qj
0
0
0
0
-1
0
0
0
0
0
0
1
1
(7-9)
4
In matrix notation, we write
P-14
=
IQ, - A,
or
P~ D + A
= T,.
(7.10)
6. Write the fluxes entering and exiting each node. This is analogous to Kirchoff's
270
Current Law:
(yDl
(bpL -
+
<bfL
4a -
(DI -
fL
-
-IgL
<bgL -
<fL
fR -
<pL
Da
=
0,
=
0,
=
0,
0,
(gR
gR -
pR
=
0,
<bpR -
(y
=
0.
(7.11)
7. Write (7.11) in matrix form as
DpL
1
-1
0
0
0
0
0
0
0
0
0
1
-1
-1
-1
0
0
0
0
0
0
0
0
1
1
-1
0
0
0
0
0
0
0
0
1
-1
-1
0
0
0
1
0
0
0
1
1
-1
-1
0
0
0
0
0
0
0
1
0
<bgL
(7.12)
0
0
'1 'fR
0
<bgR
4
DpR
In matrix notation, we write
B4P=
<D9.
(7.13)
Note that B is the transpose of AO in (7.4). Consequently, we can eliminate
271
the
5 th
row in (7.12), as the dual of step 3. We write the result as
(bpL
1
-1
0
0
0
0
0
0
0
Di
0
0
1
-1
-1
-1
0
0
0
0
fL
0
0
0
0
1
1
-- 1
0
0
0
0
0
0
0
0
1
-1
-1
0
-1
0
0
0
0
0
0
0
1
(7.14)
0
0
(ga
0
<bgR
kvpR
In matrix notation, we write
= (
(7.15)
.
AT(
8. Finally, combine (7.10) and (7.15) to form (7.1):
Ry
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
'RPL
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
Re
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
RJ L
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
RgL
0
0
0
0
0
-1
1
0
0
4bgL
0
0
0
0
0
0
Ra
0
0
0
0
0
0
0
<I)f R
0
-1
NI
(DpL
0
(bR
0
0
0
0
0
0
0
0
0
0
RgR
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
RpR
0
0
0
0
1
1
-1
0
0
0
0
0
0
0,
0
0
0
0
0
I1
0
0
1
-1
-1
-1
0
0
0
0
0
0
0
0
0
XF2
0
0
0
0
1
1
-1
0
0
0
0
0
0
0
0
XF3
0
0
0
0
0
0
1
-1
-1
0
0
0
0
0
0
XF 4
0
-1
0
0
0
0
0
0
0
1
0
0
0
0
0
(. 6
0
0
0
0
Rl R
0
0
0
0
0
-1
0
0
<I> R
(IpR
(7.16)
272
0
0
To summarize, the matrix and vector elements from (7.1) are
[D
0
0
0
0
0
0
0
0
0
RpL
0
0
0
0
0
0
0
0
0
Ri
0
0
0
0
0
0
0
0
0
lRfL
0
0
0
0
0
0
0
0
0
RgL
0
0
0
0
0
0
0
0
0
Ra
0
0
0
0
0
0
0
0
0
lRfR
0
0
0
0
0
0
0
0
0
RgR
0
0
0
0
0
0
0
0
0
RpR
0
0
0
-1
-1
1
0
0
0
0
-1
0
0
0
0
-1
1
0
0
0
-1
1
0
0
0
0
-1
1
0
0
0
0
-1
0
0
0
0
-1
0
0
0
0
0
1
(7.18)
41
)pL (Dl 4PfL Ig
(7.19)
[1=
NI 0 0 0 0 0 0 0 0 0 0 0 0 0
(7.20)
-
sF
1
(7.17)
.
P- =
Riy
Note that 'P does not appear in the network equations because node 5 has been arbitrarily selected as the reference potential,
can readily solve (7.16) for
4
gL
and
4J5 =
0. Using linear algebra techniques, we
gR, given the assumptions of linear permeances.
9
273
We can then solve for the actuator force using (2.13). We can also solve for the
fringing field flux densities and compute the force contributions from these fields.
The flux for each fringing field path is calculated by solving (7.16) for the appropriate
fringing field flux variable, <Df. The mean flux density at the target interface, Bf, is
then calculated by dividing the fringing field flux by the cross-sectional area of the
flux tube at the target interface, Af. The force contribution from the fringing field
path is then given by 3
F- =2
=(Bf 2 Af
yo
f )2 A\
Af
2po
(7.21)
The total force on the target is then the sum of the force contributions from the
working air gaps and from the various fringing fields, expressed below as
F =FgL + FgR + FL1+...+ FfLm + FfR1 +...+
BgL 2 AgL
BgR 2 AgR
Po
yo
BfLm
2
AfLm
Po
FfRn,
BfL1 2 AfL1
(7.22)
Po
BfR1
2
A f R1 +
A BfRn 2 Af Rn
Ito
[o
In contrast, a lumped parameter model that does not include the fringing fields uses
only the first two contributions (the contributions from the working air gaps) from
the right-hand side to compute the force. Moreover, these flux densities BgL and BgR
have a greater magnitude when fringing and leakage fields are not included, because
all the flux exiting or entering the pole faces will contribute to BgL and BgR rather
than to any fringing fields.
7.3
Reluctance Calculations
The next step for solving the magnetic circuit model is to calculate reluctance values
for each of the reluctances in the circuit.
3
The reluctances can be separated into
The factor of 2 in (7.21) corresponds to the fact that the magnetic circuit only models one-half
of the total actuator.
274
four main groups: the working air gap reluctances, the core material reluctances, the
fringing field reluctances, and the leakage field reluctances.
7.3.1
Reluctance Calculations for the Air Gap and Core
Calculating the values for most of the reluctances in Figure 7-3 is straightforward.
The general linear reluctance formula is
R=
/r9poA'
(7.23)
,1
where 1 is the length of the reluctance path and A is the cross-sectional area of the
reluctance path.
Figure 7-4 shows the half-actuator with dimensions labeled. With these dimensions, reluctances 1Z,,,
, LRpR,, RgL, RgR, and Ra are calculated below. The path
length for each reluctance is taken as the approximate average length of the path the
flux will follow. The reluctances are
RpL
IZpR
RgL
RgR
Iza
=
/Ir odsb'
,
Ry
2
f
yr podj
pod,f
=-.
IprIptdaf
Note that in calculating RgL and RgR, the pole face area df
area.
275
(7.24)
(7.25)
(7.26)
(7.27)
is used as the relevant
Front view
Top view
Figure 7-4: The half-actuator with dimensions labeled. The actuator core has a depth
d, into the paper. The target has a depth da into the paper.
7.3.2
Reluctance Calculations for Fringing Fields and Leakage Fields
Calculating the fringing field and leakage field reluctances is not as straightforward
because the paths that the actual field follows can be quite complex. To aid in this
calculation, we follow the flux tube procedure outlined in [74, 59]. As Roters explains
in [74], the flux tube method consists of breaking the field up into "flux paths which
are of simple shape and still probable." A 'probable' path is one that is similar to
the true flux path, in the sense that the lumped permeance of the probable path will
be close to the lumped permeance of the true path.
Figure 7-5 shows some of the flux tubes used to represent probable flux paths for
the fringing and leakage fields.4 The subscripts indicate the following: the first letter
in the subscript indicates whether the fringing flux tube belongs to the left pole face
4
Note that not all the flux tubes are shown in this 2-D figure.
276
(L) or the right pole face (R); second and optionally third letters indicate from which
location of the given pole face the flux tube originates (L, R, F, B for left, right,
front, or back, respectively; and FL, FR, BL, BR for front left corner, front right
corner, back left corner, or back right corner, respectively). The lowercase 1 in the
coil window slot indicates the flux tube associated with the leakage flux.
I,
-P
I
Figure 7-5: Numbered flux tubes are shown representing probable flux paths for the
electromagnet.
Figure 7-6 is a figure taken from [74] showing a 3-D view of the characteristic
shapes of some of the flux tubes used in the flux tube analysis. In the figure, the
faces marked A and B correspond to the pole face sides in Figure 7-5.
Surface D
corresponds to the underside of the electromagnet target. Flux tubes in Figure 75 correspond to flux tubes in Figure 7-6 in the following way: path 2 corresponds
to path 11; path 3 corresponds to path 12; path 4 corresponds to path 13; path 5
corresponds to path 14.
The flux tubes denoted by path 1 in the Figure 7-5 represent the working air gap
flux, the reluctance of which has already been calculated in (7.26). Note also that this
reluctance is not part of the fringing field, and so will not figure into the calculations
277
A11
Figure 7-6: Figure from [74 showing the 3-dimensional shapes of the flux tubes.
for RfL and
lZfR.
In the following discussion, please see [74] for more details on how
the reluctance values are derived from the corresponding flux tube shapes.
Flux path 2 reluctance
The flux tubes associated with path 2 in Figure 7-5 are shown in more detail in
Figure 7-7. The characteristic dimensions of flux tubes
g and
f.
The characteristic dimensions of flux tubes
2
2
LB, 2 LF, 2 RB,
LL, 2 LR,
and
2
and
2
RF are
RL are g and d.,.
In both cases, g figures in both the characteristic length and the characteristic area,
resulting in a cancelation in the final reluctance calculations.
Since the flux tubes represent reluctances in parallel, we can find the permeance
of each flux tube and sum them to find the total path 2 permeance associated with
each pole face. Inverting this total permeance will give us the lumped total path 2
reluctance associated with the pole face. For the total path 2 permeance of the left
pole face and right pole face, we have, respectively,
P2L
=
P2LL + P2LR + P2LF + P2LB = 2 x 0.52pof + 2 x 0.52pod 8,
P2R
-
P2RL +
P2RF - P2RB
= 2 x 0.52pof + 0.52/iod.
278
(7.28)
(7.29)
KI
Front view
Top view
Figure 7-7: Flux paths 2 and 4 showing characteristic dimensions.
Then we can write
1
7Z2L
7
(7.30)
P2L'
1
Z2R
(7.31)
P2R
Flux path 3 reluctance
The flux tubes associated with path 3 are shown in Figure 7-8. The characteristic
3
3
dimensions of flux tubes LL and RL are g, ds, and q. The characteristic dimensions
of 3 LR are g, d, and p (p has not yet been determined). The characteristic dimensions
of 3 LF,
3
LB,
3
RF, and 3 RB are g, f, and s. The flux tube permeance of path 3 for the
279
left pole face is
P3LL + P3LR -- P3LF +P3LB,
In 1
q
9
)
+
2pod_
+ 2pod, In
7r
1
+
7r
In I +
)
=
+
P3L
,
(7.32)
and the flux tube permeance of path 3 for the right pole face is
=
P3RL +P3RF +P3RB,
2pod, In
-
1Ir
3LL
LL
(1
+ Pg)
+
7r
In
(1
g
.
+
P3R
LR
9
*-q
LR
Front view
Top view
Figure 7-8: Flux paths 3 and 5 showing characteristic dimensions.
280
(7.33)
Flux path 4 reluctance
For the flux tubes associated with path 4, refer back to Figure 7-7. The only characteristic dimension associated with this flux tube is g since the mean reluctance path
2
length and the mean reluctance path area are proportional to g and g , respectively.
Therefore, the path 4 permeances for the left and right pole faces are, respectively,
P4L
P4R
=
+
P4LBL
+
P4LBR
P4RBL
+
P4RFL =
P4LFL
+
P4LFR =
4 x 0.308pog,
2 x 0.308og.
(7.34)
(7.35)
Flux path 5 reluctance
For the flux tubes associated with path 5, refer back to Figure 7-8. For paths 5LBL
and
5
LFL,
the characteristic dimensions are g and x, where x = min(q, s). In the
figure, s is shown to be the smaller dimension, and so x = s in this particular case.
For paths
5
LBR, 5LFR, 5RBL,
and 5RFL, the characteristic dimensions are g and y,
where y = min(p, s). In the figure, y = p. The path 5 permeance for the left pole
face is
P5LBL + P5LFL + P5LBR + P5LFR,
P5L
=
2 x 0.5po(x - g) + 2 x 0.5po(y - g),
(7.36)
and the path 5 permeance for the right pole face is
P4R
+
P5RFL,
=
P5RBL
=
2 x 0.5pto(y - g).
281
(7.37)
Leakage reluctance
The only remaining reluctance to be calculated is the leakage reluctance between the
left and right poles. This flux tube is shown in Figure 7-9. A typical leakage flux
path is shown in the dotted blue line, while a typical flux path through flux tube 3 is
shown in the dotted black line.
h
Front view
Top view
Figure 7-9: The leakage flux tube showing characteristic dimensions.
The dimension w -
f
is the reluctance path length, and d, and k are used to
calculate the reluctance path area. The leakage reluctance is then given by
w- f
(7.38)
1-odsk
It remains to find k. The dimension k is chosen such that the black dotted path
in Figure 7-9 is equal in length to that of the blue dotted line, w -
f,
the leakage
reluctance path length. The reasoning for this is that flux will want to flow through
282
the lowest reluctance path available. If the black dotted path is longer than w -
f,
the flux will be more likely to travel through the blue dotted path instead. If the
black dotted path is shorter than w -
f,
the flux will be more likely to travel from the
left pole through the left black dotted path, then through the armature, then through
the right black dotted path, into the right pole. Thus, we set the length of the black
dotted path (denoted 13) equal to that of the blue dotted path (denoted 11) to give an
approximate boundary between flux tubes 3 and the leakage reluctance path,
S27r
(g + h - k) X2=w-f
4
=
11.
(7.39)
Solving for k yields
k
=
h7r
f -Wg.
(7.40)
Substituting this value for k into (7.38), we can write R, as
(
R,=
-
(7.41)
-
-
pod, (h - w-f _
Now that k is determined, we can calculate p (see Figure 7-8) for (7.32) and (7.33),
p=h-k
7.4
+g=
W- f
.
(7.42)
Solving the Magnetic Circuit via Nonlinear
Network Theory
In the real magnetic circuit, the reluctances will not all be linear. That is, all reluctances associated with the core material will have nonlinear B-H characteristics. In
Figure 7-3, these reluctances are RY, RpL,
RPR,
and IZa. Consequently, if we want to
capture the nonlinear behavior of the circuit, the linear network theory presented in
the previous section will not suffice. Instead, we turn to a procedure outlined in [24]
283
called Tableau analysis. This method is capable of handling such nonlinear circuits.
We present this analysis by way of example for the circuit in Figure 7-10. This
circuit is identical to that in Figure 7-3, except we have three more variables labeled:
To, the potential at the node between the source and RY, <D., the flux associated
with the flux through the MMF source, and TP, the potential drop associated with
the source. As with the linear analysis, we assume only two fringing field reluctances
(RfL and
RJR)
The analysis can be extended to incorporate additional
for now.
fringing field reluctances.
T
pL
fZL~i
P
I
PL
]7J2
jJFii2
j4
JR1
1R2
opR
1pR
]ZLm
..
L
j'kLm
3
1?1
To6
g;L
..
P'J I
5
|ZI
fDRn
gR
.
'
.
'
11
Figure 7-10: Magnetic circuit for example Tableau analysis.
The steps that comprise the Tableau analysis are as follows:
1. Write the fluxes entering and exiting each node. This is the same as Step 6 in
284
Section 7.2, except we must also include the flux entering and exiting node 0:
(Ps - (DY
vL
Yp 'DpL
4DI
-
(DfL
-
L +
a
-
-
'IgR
(fR -
-
0,
(DgL
gL-(a
4fL -
-
=0,
=0,
0,
(gR
)pR
=0,
= 0.
(7.43)
In matrix form, after eliminating the equation associated with node 5, we write
1
-1
0
0
0
0
0
0
0
0
0
1
-1
0
0
0
0
0
0
0
0
0
1
-1
-1
-1
0
0
0
0
0
0
0
0
1
1
-1
0
0
0
0
0
0
0
0
0
1
-1
-1
0
-1
0
0
0
0
0
0
0
0
1
(DpL
0
4
)fL
0
0
4bgL
.(7.44)
0
0
0
DgR
DpR
In matrix notation, this is equivalent to
AT4 = 0.
(7.45)
Note that A here is not identical to A in the linear analysis: A in the nonlinear
analysis has an additional column corresponding to node 0.
2. Write the MMF drop across each reluctance as a function of node MMF's. This
285
-
is analogous to Steps 1 and 2 in the linear analysis:
Ts
=
4I y
--
I
=
qO,
- IF,,
1 qf2
2,
5,
-F
TfL
q2
T gL
T2 -
3)
T3 -
4,
XF4 --
W5
T4 -
5,
1P
=
TfR
1
X gR
(7.46)
-
=To
L
'J
'F
TJ pR
A 3,
=
6
In matrix form, after the column associated with node 5 has been eliminated,
we write
F;
1 0
0
0
0
-1
0
qpL
0
-1
1
0
0
0
0
IF
0
0
-1
0
0
0
[
0
T2
f
0
0
-1
1
0
0
0
qfg
0
0
-1
1
0
0
0
0
0
0
-1
1
0
0
Tf
0
0
0
0
-1
0
0
'ngR
0
0
0
0
-1
0
0
0
0
0
0
1
,Q, + ATF
0.
'Ta
j pR
-
(7.47)
0
0
In matrix notation, we write
286
(7.48)
Here also, 1@ and AF are not identical to the similarly-named vectors in the
linear analysis: in the nonlinear analysis, these vectors include the elements XF,
and To, respectively, which are absent from the linear analysis.
3. Write the constitutive relations for each reluctance and for each MMF or flux
source. This is similar to Step 4 in the linear analysis, except we include sources
as well as reluctances.
The nonlinear reluctances are characterized by the B-H relationship of the
ferromagnetic material. In order to manipulate the B-H relationships to involve
the circuit variables, Tj and
Dj5,
we note first that Ii is related to Hi by
Tj = Hili, where 1i is the length of the reluctance path of interest. We also
note that oi is related to Bi by 4Di = BjAj, where Ai is the cross-sectional area
of the reluctance path of interest. Writing the B-H relationship as B(H), we
can then express the general form of the constitutive relation for a nonlinear
reluctance as
Di= AjBj = B (Hj) = B
,(7.49)
where B(Wi/li) denotes the B-H relationship in terms of 'i.
The exact B-H
relationship will depend on the particular ferromagnetic material used in the
actuator. It can include hysteresis; however, for the simulation results presented
later in this chapter, we use a nonlinear function that includes saturation but
does not include hysteresis.
For each constitutive relationship, we subtract the quantity on the right-hand
side from both sides so that the resulting quantity on the left-hand side is equal
to zero. The parameters AY, ApL, Aa, and ApR are the cross-sectional areas
associated with the core material reluctances.
5
For our magnetic circuit, we
The subscript i is a generic notation meant to signify any of the core material reluctance subscripts.
287
then write the constitutive relations as
fi(Ts) = Ts + NI = 0,
f(
4) =
f3 (IfpL i'pL) =pL
'PI
f4('J!141)
f7 (
T gL)
a,
fs(JfR,
0,
0,
RgL4gL
0,
gL-
)=
a - AaB
fR)
fR
pR) =
0,
'FfL - RfLTfL
-
f9(XIgR, 4DgR) = TgR -
foQ1J Ri
=0,
ApLB
-
- R4 1 =
fA('F fL, (fL)
f6('JgL,
(
y - AyB
4pR
-
= 0,
fR
0,
RgR gR
0,
RfR
ApRB
h
=
0.
(7.50)
(7.45),(7.48), and (7.50), written in matrix notation as
ATD
qlr - Ax
f(Air,4D)
This system consists of (n - 1)
=
0
=
0
=
0.
(7.51)
+ 2b nonlinear algebraic equations in (n - 1) + 2b
variables, where n is the number of nodes in the circuit and b is the number
of branches in the circuit (one branch for each reluctance element and source
element). We can solve this system of nonlinear equations using MATLAB's
command 'fsolve' for instance.
288
7.5
Magnetic Circuit Simulations
In this section, we present simulation results using both the linear magnetic circuit
and the nonlinear magnetic circuit. We also present simulation results using Finite
Element Analysis (FEA). We compare the simulated actuator force to the experimentally measured force in lab. Matlab scripts were written to run the linear magnetic
circuit simulations. For the nonlinear magnetic circuit simulations, we used a Matlab
script initially and then switched to using Spice, a circuit simulation software.
7.5.1
Experimental Setup
To measure the actuator force, we used the test fixture shown in Figure 7-11. This
test fixture was designed and built by Tony Poovey. More details on its design can be
found in Poovey's Master's thesis [71]. The test fixture consists of a platen mounted
on three load cells with kinematic mounts. An electromagnetic target is bolted to
the underside of the platen. The load cells are mounted to three micrometers. These
micrometers are used to set the air gap. Three capacitance probes measure the gap.
Inside the test fixture cavity is an electromagnet stator that generates a force on
the platen. Using this fixture, we can map the force-gap-current relationship for the
actuator.
The kinematic mounts are silicon-carbide and the force is measured by
quartz load cells, which provide for a high-stiffness force loop. The first mechanical
resonance of the structure is above of 1 kHz. A sample electromagnetic actuator that
can be used with the test fixture is shown in Figure 7-12. The particular actuator
shown has a silicon-iron core.
7.5.2
Linear Simulations
Figure 7-13 shows the experimentally measured force compared to the simulated force
for a 50 pim gap, where we used the linear magnetic circuit to simulate the actuator.
The linear simulation was run by increasing current until the flux density in the core
reached its saturation value. The simulation shows a good fit at low currents, but
departs from the measured data at high currents. A nonlinear simulation is needed
289
Figure 7-11: Test fixture for measuring electromagnetic actuator characteristics.
Figure 7-12: Sample electromagnet for testing.
to model the high-current behavior. For comparison, we also show a simulation that
did not include the force contributions from fringing fields. We see that at a 50 jim
290
gap, the fringing fields have very little influence.
F v. I
250
200-
150-
- 100measured
sim w/ fringing field contributions
sim w/o fringing field contributions
50
04
0
0.5
1
1.5
current [A]
Figure 7-13: Simulated and measured force at 50 pm gap. Linear magnetic circuit
used for simulation.
Figure 7-14 shows the experimentally measured force compared to the simulated
force for a 450 pim gap. The power amplifier used in the experiments was only capable
of providing up to 1.3 A, so force data up to saturation was not available.
The
simulated force does not fit quite as well with the measured force as it did at the
50pm gap.
This may be because the leakage flux dominates more at the larger
gaps, so we may need to use a more spatially-distributed MMF source and multiple
leakage reluctances to capture the leakage flux behavior more accurately. Moreover,
the fringing fields dominate more at larger air gaps, so any inaccuracies in the flux
tubes will have a more pronounced effect. However, we see that at this larger gap,
including the fringing field contributions to the force generates a noticeably better fit
than not including them.
7.5.3
Nonlinear Simulations
To provide a better model fit at high current levels, a nonlinear saturation model for
the permeable material reluctances was developed. B-H measurements from a 5050% NiFe toroidal core with tape-wound laminations of 50 pm thickness were used to
create an empirically-based model. Figure 7-15 shows the measured data. The data
291
F v. I
25
20 -
measured
sim w/ fringing field contributions
sim w/o fringing field contributions
-15
(D
010
504
0
0.2
0.4
0.6
0.8
current [A]
1
1.2
Figure 7-14: Simulated and measured force at 450 pm gap. Linear magnetic circuit
used for simulation.
was obtained by exciting the drive coil with a 1 Hz voltage sine wave and measuring
the current through the drive coil (proportional to H) and measuring the voltage on
a secondary coil (proportional to dB/dt) and integrating it.
B v. H
1.5
1 -;r0.5
E 0x
-0.5
-1-1.5
-40
-20
0
20
magnetic field [A/m]
40
Figure 7-15: Experimentally measured B-H data for a 50-50% NiFe toroidal core.
Two vectors were created from the measured B-H data, one to represent the H
data, and the other to represent the corresponding B data. These vectors represent
the B(H) function of the core material. Since we did not consider hysteresis nonlinearities in our magnetic circuit model at this point, we took an average between the
two branches of the major B-H loop. The result of this method is shown graphically
292
in Figure 7-16.
B v. H
1.5
CD
-
E 0.5
S0
-0.5
-1 -
-40
measured
+ nonli near function
-20
0
20
40
magnetic field [A/m]
Figure 7-16: Single-valued B-H function (red) computed from measured B-H (blue).
This single-valued B(H) function was then applied to all the core material reluctances in our magnetic circuit model, namely, RpL, RpR, 1Z, and 7Za. This results in
a system of nonlinear equations as described in Section 7.4. Once this system of nonlinear equations is set up, a nonlinear solver, such as MATLAB's 'fsolve' command,
is used to solve the circuit for a given current input (NI) trajectory.
Figure 7-17 shows a nonlinear simulation compared with measured data for a
50 lim gap. The simulation still does not accurately capture the actuator behavior
at high currents. To rectify this situation, we posited that a more complex circuit
network with additional reluctances and a distributed MMF source may be needed.
Such a network with these additional elements will be better able to capture the fact
that certain parts of the core will saturate earlier than others, which may lead to a
more gradual transition to saturation as seen in the measured data.
7.5.4
Nonlinear Simulations with Additional Lumped-parameter
Elements
In an effort to improve the accuracy of the magnetic circuit simulation, we augmented
the magnetic circuit with additional elements. The modified circuit is shown in Fig-
293
F v. I
250
200
[
150
0 100
50 F
go0
measured
nonlinear simulatio n
0.5
1
1.5
current [A]
Figure 7-17: Simulated and measured force at 50 jrm gap. Simulated force includes
nonlinear core reluctance elements.
ure 7-18.
We now have two MMF sources with the total amp-turns split evenly
between the two. This better reflects the fact that the MMF from the actuator core
is spatially distributed. We have also split the leakage reluctance into two separate
reluctances.
This is intended to capture the phenomenon that leakage flux will be
concentrated near the bottom of the coil window because the MMF drop across the
coil window is greater here.
Figure 7-18: Augmented magnetic circuit.
294
For simulations with the modified circuit, we used the circuit software SPICE.
Figure 7-19 shows a nonlinear simulation of the augmented circuit compared to measured data for a 50 pm gap. We have also included the nonlinear simulation using
the original circuit. We see very little difference between the two simulations. The
additional circuit elements did not seem to help much. As we will discuss later in
this section, it turned out that the reason for the discrepancy is because the actuator
core material was not identical to the toroid core material from which we obtained
the B-H data for our magnetic circuit model.
F v.I
250
200 -
-
0o
150
10050
-
0
-
0
0.2
0.4
0.6
Experimental
Original nonlinear sim
Augmented nonlinear sim
0.8
1
1.2
1.4
current [A]
Figure 7-19: Simulated and measured force at 50 pm gap. Augmented nonlinear
simulation (green) uses additional lumped parameters in model.
7.5.5
Nonlinear Simulations Using Finite Element Analysis
Since we seemed to be reaching the point of diminishing returns by adding more
lumped parameter elements to the nonlinear magnetic circuit (the material discrepancy had not been discovered at this point), we switched our efforts to finite element
modeling of the actuator. We used COMSOL Multyphysics software [28] for this analysis. COMSOL permits the user to enter nonlinear B-H functions for the material
properties. Figure 7-20 shows a 2-D FEA model of the actuator under test. We used
the data from Figure 7-16 for the material B-H relationship.
295
Figure 7-20: COMSOL FEA model of actuator.
Figure 7-21 shows the FEA simulated B-I curve compared to measured data for
a 50 pm gap. We see a similar divergence of the simulated data from the measured
data that we saw with the nonlinear magnetic circuit simulations. We also note that
at saturation, the FEA had difficulty solving the model. As we discuss in the next
subsection, the reason for this is because of the material discrepancy.
F v. I
400
FEA unable to solve
300
after this point
z
(U
200
_
10
0
0
0.2
0.4
0.6
-
Experimental
-
FEA nonlinear simulation
0.8
1
1.2
1.4
current [A]
Figure 7-21: FEA simulation and measured force for 50% NiFe at 50 pim gap.
296
7.5.6
Finite Elemenent Analysis of 49% NiFe
We later discovered that the actuators we had been testing used 49% NiFe cores
rather than 50% NiFe cores [58]. In our simulation models, we had used data from
two
a 50% NiFe core. It turns out that there is a significant divergence between the
materials at higher magnetic fields: 50% NiFe has a much steeper B-H curve than
49% NiFe. We then measured the B-H curve on a 49% NiFe sample with laminations
is
of 25 pim thickness. The sample is shown in Figure 7-22. The measured B-H curve
shown in Figure 7-23.
10.8 cm
Figure 7-22: 49% NiFe sample used to measure B-H curve.
If we compare this graph to the 50% NiFe B-H curve from Figure 7-16, we see
that B-H curve from the latter transitions to saturation much more rapidly. This
is confirmed by B-H data from Vacuumschmelze [151, replotted below on a log-log
plot. While the 49% NiFe has higher initial permeability, 50% NiFe has a steeper
transition to saturation and reaches saturation first. Because of this, using 50% NiFe
for the actuator cores could result in a more efficient actuator. However, because of
the accuracy requirements of the lithography application, 49% NiFe is likely to be
the better option for our purposes because the gentler transition to saturation will
be easier to manage. Moreover, as we discuss in Chapter 9, eddy currents affect 50%
297
B v. H
1
-
-
1.5
x
5 -0.5-1
-
-1.5
-300
-200
-100
0
100
magnetic field [A/m]
200
300
Figure 7-23: Experimentally measured B-H data for a 49% NiFe toroidal core.
NiFe at very low frequencies, complicating actuator linearization further. The 50%
NiFe is typically used for switching applications rather than for actuator applications.
101
100
10-1
0)
:3
x
10-2
ji
---
102
102
100
49% NiFe
. 50% NiFe
1
4
magnetic field [A/mi
Figure 7-24: B-H data from Vacuumschmelze for 49% NiFe and 50% NiFe.
We used COMSOL software to run a 2-D FEA of the actuator, now using the
nonlinear B-H data for 49% NiFe. Figure 7-25 shows the FEA simulated force of the
actuator under test compared to measured data for a 50 jim gap. We see that the fit
is much better than the magnetic circuit simulations that used 50% NiFe data.
298
Fv.I
250
0
-Experimental
--
0
0.2
0.4
0.8
0.6
current [A]
FEA
1
1.2
Figure 7-25: FEA and measured force at 50 pm gap. Simulated force used 49% NiFe
B-H data.
7.6
Summary
We developed linear and nonlinear magnetic circuit techniques to model a reluctance
actuator. We simulated these models and compared them to experimental measurements on an NiFe actuator we tested in lab. We also modeled the actuator using
FEA. The models showed good agreement at low current levels but divergence at
high current levels. This was because we had been simulating 50% NiFe while the
actuators that were tested were 49% NiFe. From testing toroidal samples of the two
materials, we learned that while the saturation flux density between the two materials is similar, there is a significant divergence between the two in how rapidly they
transition to saturation. The FEA simulation showed much better agreement once
we changed the material properties to those of 49% NiFe. We concluded that 49%
NiFe is likely better for our actuator prototype.
In the nonlinear magnetic circuit simulations, the nonlinear elements representing
the core reluctances were represented by single-valued functions. These reluctances
could be replaced by hysteretic elements using the models developed in Chapter 3,
but this has not been addressed in this thesis. This would allow the magnetic circuit
model to capture the hysteretic behavior of the actuator.
299
We can adapt the magnetic circuit techniques developed in this chapter as a
design tool used to size actuators.
In the next chapter, we present a design for a
1-DoF testbed for testing reluctance actuators. We use the models presented in this
chapter to size the reluctance actuator prototype.
300
Chapter 8
One-DoF Reluctance Actuator
Testbed Design
In this chapter, we present the design of a 1-DoF testbed for testing a reluctance
actuator prototype with flux feedback control. The testbed includes a moving mass
on an air bearing setup and a high-resolution linear encoder for position sensing and
control.
Load cells provide a force measurement of the actuator.
The reluctance
actuator incorporates a sense coil for flux sensing and control.
We first present several alternative concepts we considered for testing reluctance
actuator force accuracy. We then give an overview of the design of the 1-DoF testbed.
Finally, we present the detailed design of a reluctance actuator prototype for use with
the testbed.
8.1
Design Requirements
We require a design that can measure the force accuracy of a reluctance actuator
prototype when a dynamic force profile is applied.
We also require that we can
change the actuator air gap dynamically so that we can evaluate the force accuracy
of the reluctance actuator in the presence of a gap disturbance. Finally, we require a
design that permits us to test the accuracy of our hysteresis models and flux control
schemes. The following is a list of components and capabilities for the testbed needed
301
to satisfy these high-level requirements:
"
Actuator force measurement
" Dynamic motion capability
" Position measurement
" Actuator flux measurement
" Actuator current measurement
" Real-time operating system for control
" Amplifiers to drive actuators
8.2
One-DoF Testbed Concepts
We present in this section several general concepts for testing the reluctance actuator
force accuracy. We focused on 1-DoF designs. This constrains the design space and
permits for relatively fast turnaround for the design and assembly phases. Thus, this
actuator testbed cannot be used to explore effects of rotation, which will be relevant
to the 6-DoF levitation case. While investigating off-axis effects is critically important
for scanning lithography, the focus of this thesis is on modeling and control of the
drive axis.
Roberto Melendez has investigated off-axis behavior of the reluctance
actuator in his thesis [62].
For the main motion stage, we focused on concepts that used a linear air bearing.
The reason for this is that an air bearing closely approximates the near-zero stiffness in
the scan axis of the magnetically-levitated short-stroke stage in a lithography machine.
This allows us to isolate the stiffness of the reluctance actuator from other sources of
stiffness. Linear roller guides have friction and also exhibit non-zero stiffness for small
motions. Flexures stages also have non-zero stiffness. Another reason for focusing on
air bearing designs is that we had a linear air bearing in lab that we could adapt to
our design requirements.
302
8.2.1
Differential-mode Actuation Configuration
Figure 8-1 is a 1-DoF testbed concept in which the moving mass is actuated differentially. The idea is that a common-mode force can be applied to the actuator on either
side of the payload, thereby nulling the net force on the payload. The advantage of
such a configuration is that we can utilize the full range of the reluctance actuator
force capability without the actuated mass accelerating and traveling long distances.
Since the payload will remain nominally at the same position, this obviates the need
for a long-stroke stage.
The position deviation from zero can be used to measure
the force accuracy of the reluctance actuator. Load cells are mounted on the stator
side of the actuators for force measurement. Locating them on the stator side avoids
measurement errors resulting from acceleration of the payload. (Note we assume zero
acceleration of the support structure.) A linear encoder is included on the air bearing
payload for metrology. One actuator can be used to generate a gap disturbance while
the other tries to maintain the desired force accuracy. One downside of this concept
is that force control and position control cannot be maintained simultaneously using
the same actuator if we want to apply a gap disturbance.
Linear encoder
Reluctance
actuator
Reluctance
actuator
ULoad cell
Load cell
Figure 8-1: Testbed concept with differential-mode actuation.
303
8.2.2
Differential-mode Actuation Configuration with Gap
Disturbance Stage
Figure 8-2 shows one of the reluctance actuator stators mounted on its own stage.
This allows a gap disturbance to be directly applied to the stator, and in this way
it emulates the long-stroke stage. This configuration is more true to the scanning
system as now the reluctance actuator can be used for positioning control of the air
bearing payload even while a gap disturbance is being applied to it.
Capacitance
probes
Piezo
actuator
Reluctance
actuator
Linear encoder
Lorentz
actuator
Gap disturbance
stage
Figure 8-2: Testbed concept with differential-mode actuation and gap disturbance
stage.
The gap disturbance stage actuator is shown as a piezo actuator. An advantage
of a piezo stage is that piezo actuators are capable of high force and high stiffness and
can therefore overcome the reaction force from the reluctance actuator stator. Other
actuators are also possible, however.
Gap metrology is shown being provided by
capacitance probes. The gap disturbance stage can be mounted on linear guides for
instance, since low friction and low stiffness are not as critical for the gap disturbance
stage. Other possibilities include a flexure stage or another air bearing stage.
A modification of this idea is to use multiple piezo actuators to provide not only
a gap disturbance but disturbances in other degrees of freedom as well. This would
permit us to investigate the off-axis behavior of the reluctance actuator.
One downside of adding a gap disturbance stage is that now there is no direct
way to measure the force of the reluctance actuator, since load cells mounted to the
304
stator will be affected by the gap disturbance stage acceleration. To rectify this, the
other end of the air bearing slide is shown with a Lorentz actuator attached. When
the air bearing slide is operated in nulling mode, the Lorentz actuator can provide
another force measurement, since the Lorentz actuator current is proportional to the
force applied by the actuator, assuming that the central mass is not moving.
8.2.3
Long-stroke Stage Configuration
Another potential configuration uses a long-stroke stage instead of a differential-mode
configuration. Such a concept is shown in Figure 8-3. The main advantage here is
that we can now command both the desired force profile to the reluctance actuator
and the desired position profile. Since demonstrating accurate servo performance is
the ultimate goal of using reluctance actuation for scanning lithography, this concept
provides greater capabilities for testing the reluctance actuator. The primary downside is much greater complexity, cost, and testbed size requirements. A larger area is
needed for the testbed since now the stage will be traveling large distances when the
forces are applied. Another downside is that the large accelerations will introduce
large reaction forces into the force frame. This is why a large reaction mass is shown
in the concept. A drift motor is included to keep the position of the reaction mass
centered. This reaction mass and drift motor increase the design complexity further.
8.3
Testbed Design
Ultimately, we chose the differential-mode configuration. It is the simplest configuration that still permits us to test the reluctance actuator force accuracy in the presence
of a gap disturbance.
A CAD model of the testbed is shown in Figure 8-4 with a reluctance actuator
on one end of the air bearing and a voice coil actuator (Lorentz actuator) on the
other end of the air bearing. A photograph of the completed assembly is shown in
Figure 8-5.
We designed the testbed such that we could interchange a reluctance
actuator with a voice coil actuator if so desired. The design of the testbed was a
305
Linear encoder (relative)
R luctance actuator
Lorentz actuator
Linear encoder ---
/
Linear motor
for long stroke
Figure 8-3: Testbed concept with long-stroke stage.
collaboration with Darya Amin-Shahidi. Many of the design details can be found in
his thesis [4]. We here give a general overview of the design.
Encoder
Mount
,
Load Cell
+
Reluctance
Actuator Stator
Reluctance
Actuator Target
Airbearing
Stator
Linear Lorentz
/Actuator
I
K
/
/
Airbearing
Shaft
Figure 8-4: CAD model of 1-DoF testbed.
Figure 8-6 shows a cross-sectional view of the CAD model of the testbed. The
testbed motion stage uses a linear air bearing from New Way [65].
Metrology is
provided by a high-resolution linear encoder that we had custom-made by Sony [78]
306
Figure 8-5: Photograph of 1-DoF testbed.
to fit with our air bearing. It is adapted from their BH20 model to use a short scale
with a 7 mm length and with 1 pm grating pitch. It is capable of 0.25 nm resolution
at 1000-times interpolation. The scale is mounted to the air bearing slide, and the
encoder read head is mounted to the air bearing stator. In order for the air bearing
to accommodate the encoder, we had to modify the air bearing. For details of these
modifications, see [4].
Electromagnet
Electromagnet
Stator
Air bearing
Target
Voice Coil
Motor
Load Cell
Encoder
Load Cell
Figure 8-6: CAD model of cross-sectional view of 1-DoF testbed.
A reluctance actuator target is bolted to one side of the air bearing slide. The
reluctance actuator stator is connected by three Kistler 9212 load cells [52] to the
support block in the stationary frame. These load cells provide high-stiffness force
measurements of the actuator force.
However, because they are quartz piezo load
307
cells, they are AC-coupled and cannot measure a true static force. The three load
cell outputs are tied together and input to a single Kistler 5010 charge amplifier. In
this way, we obtain a sum of the three load cell charge outputs, proportional to the
total actuator force.
On the other end of the air bearing is shown a voice coil linear actuator. This
provides the nulling force that opposes the reluctance actuator force.
be used to generate a gap disturbance to the stage.
BEI Kimco model LA30-46-000A [10].
It can also
The voice coil actuator is a
Its specification sheet lists it as capable of
generating a peak force of 445 N at 12.7 A. One reason a voice coil actuator was chosen
is because the current is close to linear with the force generated. If we measure the
voice coil current accurately, this gives us an estimate of the force. If the air bearing
armature can be approximated as a pure mass, which is valid for low frequencies, the
voice coil actuator gives us another measure of the reluctance actuator force when
the position is nulled. Unlike the load cells, the voice coil can also give us a DC
measurement of force.
Figure 8-7 shows a system diagram of the testbed. The encoder signal is sent to
the interpolator and then brought into the dSPACE real-time controller (RTC) [35]
via the digital I/O ports. The charge amplifier output for the load cells is brought
into the RTC via an A/D input. The RTC sends a voltage drive signal to the power
amplifier via dSPACE's D/A outputs to drive the actuators. Different amplifiers were
used at different phases of testing. Kepco bipolar operational amplifiers (Model BOP
36-12) [50] were used for both the reluctance actuator and the voice coil actuator when
the testbed setup was in Wilton, CT at ASML. When the setup was brought back
to MIT, both actuators were driven with a dual-channel Crown DC300A amplifier
[29]. Sense resistors are used to measure the actuator currents. These signals are sent
to Tektronix AMD502 differential amplifier channels [80] to reduce common-mode
noise before being passed into the dSPACE RTC. The signal from a sense coil on the
reluctance actuator for estimating magnetic flux is sent to a separate AMD502 before
being sent to the RTC.
308
dSpace
RTC
Interpolator
Enc Count
B1100HC
AM502
Buffer
dAjdt
BH20
Encoder
Buffe
(Sensed Linkage)
Kistler
9212
4.-
Charge
-
-
F
Amplifier
Kistler 5010
Is (Sensed Current)
----
-
-
Fpower
Air bearing
Reluctance
Actuator
IAmplifier
Figure 8-7: Testbed system diagram.
8.4
Reluctance Actuator Prototype Design
A photograph of the reluctance actuator prototype stator is shown in Figure 8-8.
The reluctance actuator prototype was designed to be a scaled version of a fullsize actuator.
Each actuator designed for a lithography machine requires -1500 N
maximum force (assuming two actuators per side for the short-stroke stage).
We
designed our prototype to have a maximum force capability of roughly 25% of this,
or about 300-400 N. This decision was made for several reasons.
First, the voice
coil actuator required can be much smaller; this adds less weight to the assembly
and takes up less space. Second, the cut cores used for the reluctance actuator are
generally less expensive if they are smaller. Finally, power management and amplifier
selection are much more feasible when we do not have to worry about large voltages
and currents resulting from high-force and high-inductance actuators.
We chose to design a three-pole actuator with a single actuator coil wound around
the center pole face. One advantage of a three-pole actuator over a two-pole actuator is that only a single actuator coil is needed.
A two-pole actuator normally
would require two coils for the purposes of symmetry. However, a real device should
probably use a two-pole actuator because it is less liable to parasitic moments (see
309
Support Block
Load cells
Stator core
Stator housing
Sense coil
Figure 8-8: Reluctance actuator stator prototype.
Section 2.6.2).
The target can also be made thinner for the same force capability
with a three-pole actuator. The actuator stator consists of two tape-wound cut cores
adjacent to each other as was shown in Figure 2-15. For the core material, we chose
SuperPerm49 from Magnetic Metals. This is a nickel-iron alloy with 49% nickel. We
chose NiFe over CoFe because it was less expensive. However, because CoFe has a
higher saturation flux density and can generate higher forces, CoFe is probably the
correct long-term solution.
The cut cores are laminated with 0.004"-thick (~100 rim) laminations. Magnetic
Metals lists the stacking factor as 0.9.
AWG
#
The actuator coil consists of 280 turns of
21 magnet wire. The coil resistance is R = 1.5 Q and the coil inductance at
g = 500 pm is 46 mH. In the rest of this section, we present a more detailed analysis
of the design procedure. The general procedure can be outlined as follows:
1. Size the actuator pole face area for the desired maximum force.
2. Determine number of amp-turns required to achieve maximum force at largest
expected gap.
3. Size coil window area to prevent coil from overheating.
310
4. Determine number of turns based on constraints such as peak current.
5. Select wire gauge such that desired number of turns are met.
6. Select cut cores and I-bar with dimensions close to those needed for required
pole face area and coil window area.
7. Iterate on design to finalize amp-turns and wire gauge.
8. Design housings for stator and target.
8.4.1
Actuator Pole Face Sizing
The first step in designing a reluctance actuator is to size the pole face area for the
desired force. As a first iteration, we can use (2.13) to size the total pole face area
A. We replace F with the maximum desired force, Fmax, and Bg with the maximum
flux density, Bmax, and write
A =B2poFmax
B2 a
(8.1)
The variable Bmax is limited by the saturation flux density, B, of the core material.
The average flux density in the air gap will be further limited by the lamination
stacking factor, which decreases the Bm,,x by a factor of 0.9. We include an additional
factor of 0.9 so as to bias toward a conservative design. The saturation flux density
is listed as 1.5 T by the manufacturer, so we estimate Bmax to be 1.215 T. If we set
Fmax = 400 N, then A = 681 mm2
8.4.2
Coil Amp-turns
To estimate the number of amp-turns necessary, we use (2.8) solved for NI with
B9 = B, and g = gmax. We write
NI = 2gmaxB.
AO
311
(8.2)
Our nominal operating gap go is 500 pm. A typical gap disturbance is tens of microns.
As a conservative estimate then, we set gmax = 600 pm. This results in NI = 1440 Aturns as a starting point for our design.
8.4.3
Coil Window Area
We can write the amp-turns in terms of the coil window area, A., and the average
current density, J, as
NI = JAW.
(8.3)
The coil window area is indicated by the white dotted line in the half-actuator of
Figure 8-9. A rule of thumb for current density is that an average current density of
1
X 106
A/M 2 does not require any active cooling, while an average current density
of 1 x 10 7 A/M 2 requires water cooling. We stipulate a peak current density of 1 x
107 A/M 2 at the nominal gap. This translates to 1.2 x 10 7 A/M 2 at the maximum
gap.
Provided that we maintain a low duty cycle, this should be feasible without
water cooling. With this peak current density,
JpkAW equal to the peak NI. This results in A
Jpk,
we can solve for AW by setting
= 120 mm 2 . However, this is without
taking into account the wire packing factor.
Figure 8-9: Coil window.
312
8.4.4
Wire Gauge
is well
Suppose we want to limit ourselves to roughly a peak current of 5 A. This
within the capability of several commercial amplifiers we have available in lab. From
Section 20.3 of [48], AWG
#
2
23 is listed as having a packing density of 237 turns/cm2
in our coil.
Using this wire gauge, we should be able to fit an approximate 284 turns
current
With N = 284 turns, we expect a peak current of 8.46 A at gmax and a peak
of 5 A at the maximum gap.
8.4.5
Selecting Cut Cores and I-bar
cutUsing the foregoing parameters of A and A, as our starting point, we selected
dicores from Magnetic Metals that would roughly match these parameters. The
and
mensions for the chosen cut core are shown in Figure 8-10. The dimensions A
to
G are set by the specified dimensions given in the figure. The strip width refers
the lamination width. The total pole face area (which includes two c-cores) for these
dimensions is A
726 mm2
0.75"
E = 0.375"
F = 0.625"
D
y
G
=
=1.25"
Z
Figure 8-10: Dimensions of c-core. Figure taken from Magnetic Metals website.
The dimensions of the I-bar are shown in Figure 8-11. The x- and z-dimensions
are oversized by 1/8" on each side relative to the c-core stator.
These dimensions
were oversized so that the actuator would be less sensitive to misalignment and to
any off-axis motion. The I-bar y-dimension is oversized by
1/16".
This dimension was
oversized because we wanted to ensure that target saturation was not what limited
our maximum force.
313
0.875"
STRIP WIDTH
0.437"
3.0"
1
2
X
Figure 8-11: Dimensions of I-bar.
8.4.6
Simulations of Actuator Design
Using the c-core and I-bar dimensions, we simulated the actuator design using the
linear magnetic circuit presented in Chapter 7. The simulated force is plotted against
coil current density in Figure 8-12 for a 500 pm gap. The simulated actuator reaches a
maximum force of 400 N at a current density of 7.5 x
106 A/M 2 .
This current density
is based on the original coil window area from Section 8.4.3. In reality, it will take
a higher current density to reach this force because the effect of saturation was not
taken into account in the linear simulation.
F v. J
400
2300
.....-.
.....
-.-.
--.
..-
-.
~~~-..
..
.
-..
..........
..
~
...
0200
100
0
2
4
current density [A/m 2]
6
8
x 106
Figure 8-12: Magnetic circuit simulation of force versus current density for reluctance
actuator design.
We ran a nonlinear 2-D FEA simulation of the actuator to take into account the
314
effect of saturation.
2
We also increased the coil window area to 135 mm to make
better use of the area available from the cut cores chosen.
Figure 8-13 shows the
FEA model at saturation. The stator cores are completely saturated. The target on
the other hand has several areas with very low magnetic flux density. This suggests
possibilities for optimizing the target shape: for instance, one could chamfer the top
corners as shown in Figure 8-14 in an effort to save mass. The target height could
also be reduced.
XN)m0.W
Surfmw M@a~kgcm
*
nem(T] S~hdmatik
.wtMagutc
mdvwly ,um[T]
AQO
Mam
IA
0.02
12
001
I
0
0A
0
-0.07
-OA43
0.2
-044
-0AS
404
-013
-012
-01
0
01
0A
003
0.4
0.0
Mhs I A3 I"
Figure 8-13: FEA model of reluctance actuator prototype at saturation.
Figure 8-14: Concept of target design with top corners chamfered to reduce mass.
Figure 8-15 shows the simulated force plotted against the coil current density from
2
the FEA simulation. We reach 400 N at about 6.5 x 106 A/M . There are two caveats:
1) the actual force reached will likely be somewhat lower than the simulation indicates
because the FEA model is 2-D and does not take into account the effect of fringing
in the out-of-plane dimension; 2) the peak current density will be somewhat higher
315
than indicated because the packing factor of the coil turns within the coil window
reduces the ratio of copper area to total coil window area.
Fv.
J
500
400-
300
200
10AA
00
2
4
6
8
2
current density [A/m ]
10
12
x 106
Figure 8-15: FEA simulation of force versus current density for reluctance actuator
prototype.
8.4.7
Housing Assembly Design
A CAD model of the actuator housing assembly is shown in Figure 8-16 along with
two cross-sectional views. We worked with Fred Sommerhalter, who provided inputs
in finalizing the housing design. In the picture, a PCB is shown on the face of the
stator housing. We originally intended to design our sense coil on a PCB that fit
over the pole faces. We ended up winding the coil by hand instead using thin coaxial
cable.
The stator housing includes a deep pocket to locate the c-cores and then a shallower pocket against which the coil sits. The hole in the center of the back of the
stator housing is a 1/4"-20 tapped hole used to preload the stator against the load
cells. The three counterbore through-holes are used to bolt the stator to the load
cells. The c-core pole faces extend beyond the face of the housing so that there is
room to place a sense coil after assembly.
The target housing has a pocket in which the I-bar resides. The I-bar rests against
two flats on the bottom surface of the pocket. These flats allow a space for epoxy
316
C-CorePCB
C CCil
C-Cor
Co
C-Core
-Cr
PCB
Figure 8-16: CAD model of reluctance actuator housing.
on the backside of the target. There are four counterbore clearance holes for
20 screws to bolt the target to the air bearing slide.
1/4"-
The housings were made of
aluminum for ease of manufacturing, for its high thermal conductivity, and for its
nonmagnetic properties.
8.4.8
Manufacture and Assembly
Fred Sommerhalter manufactured the housings, wound the coils, and assembled the
completed actuators for us.
Figure 8-17 is a photograph of the stator aluminum
housing, and Figure 8-18 shows two photographs of the stator with c-cores and coil
before potting. With the larger coil window area, we were able to specify a larger
wire gauge and so we selected AWG #21.
The coil consisted of 280 turns. Kapton
insulating tape [36] is placed on the aluminum housing surfaces that come into contact
with the coil and the cores.
317
Figure 8-17: Stator housing.
Figure 8-18: Actuator stator before potting.
The c-cores and coil were potted in place with EC-1012M/EH-20M potting compound [38], a thermally conductive epoxy. A type K thermocouple was also placed
inside the housing next to the coil to permit temperature measurement. Figure 8-19
shows a set of potted actuators.
After potting, the cut-core pole faces were ground flat and parallel to the back
face of the stator housing. The parallelism is important so that force applied by the
reluctance actuator is in the direction of the air bearing slide and in the direction
of the load cell sensing axes.
Before grinding, hot melt glue was applied to the
portion of the cores protruding from the housing. This was done with the intention
to support the cores during grinding so that the laminations would not 'mushroom'
outwards from the the grinding procedure.
318
A photograph of the hot glue applied
Figure 8-19: Potted actuators.
to the cut-cores is shown in Figure 8-20. Unfortunately, the c-core laminations still
mushroomed out after grinding in spite of the presence of the hot glue. Figure 8-21
shows the laminations of the completed stator mushroomed out.
Figure 8-20: Hot melt glue applied to cut cores.
For a second batch of actuators, we decided not to grind the faces of the c-cores,
as the c-core pole faces are already quite flat. The challenge then is ensuring that the
pole faces are parallel to the back surface of the stator housing. To do this, Fred built
a fixture with two sets of flat, parallel surfaces. A sketch of this fixture is shown in
319
Figure 8-21: Photograph of pole face laminations frayed outward from grinding.
Figure 8-22. The outer set of surfaces was placed against the front surface of the stator
housing. The inner surface of the fixture is located above the c-core pole faces. Two
permanent magnets are placed on the back side of the fixture to pull the cut-cores up
against the fixture. The areas marked 'adhesive' are where Kapton insulating tape is
located. Figure 8-23 shows a photograph of the stator with the fixture and permanent
magnets in place. With the coil in place and the fixture locating the cut-cores, the
cavity was filled with epoxy and then left to cure.
-752
Figure 8-22: Fixture for making c-cores parallel to housing surface.
The targets were assembled by placing the I-bars in the target housing pockets
and then using a mold for the epoxy that fit around the top surface of the I-bar.
Kapton tape was placed on the surfaces of the housing pocket to insulate it from
the target. Figure 8-24 shows the target housing and also shows the mold used for
320
Figure 8-23: Photograph of fixture for making c-cores parallel to housing surface.
potting the target assembly.
Figure 8-24: LEFT: Target housing; CENTER: Housing with I-bar in pocket; RIGHT:
Mold for potting target assembly.
Figure 8-25 shows the potting procedure.
The leftmost assembly in the figure
shows the target assembly with the mold in place and epoxy in the opening.
The
center assembly shows the target assembly after the epoxy has cured and the mold
has been removed. After the mold has been removed, the cured excess epoxy and
target face are machined down to size. This is shown in the rightmost assembly.
321
Figure 8-25: Target assembly showing potting procedure.
The target is then ground flat and parallel to the back surface of the housing. It is
more critical to grind the I-bar surface than the c-core faces because the I-bar forcegenerating surface has greater surface roughness. The c-core pole faces are formed
by cuts made in the original tape-wound core: this results in a flat surface where
the cut is. The cuts in the I-bar are at the two ends of the I-bar, where flatness is
not critical. In contrast, the force-generating face of the I-bar is formed by the edges
of the laminations as they are wound, resulting in an uneven surface. The potting
compound acts to support the laminated I-bar during grinding so that the laminations
do not mushroom out. This worked much better than the hot glue method did for the
c-core pole faces. The grinding direction may have also helped. Figure 8-26 shows a
completed target assembly.
8.4.9
Sense Coil
For the sense coil, we used miniature coaxial shielded cable. The sense coil is shown
in Figure 8-27.
We placed kapton on the epoxy surface for additional insulation
between the actuator coil and sense coil. We then placed double-sided tape on top of
322
Figure 8-26: Completed target assembly.
the kapton. This kept the sense coil in place as we wound it. After winding the coil,
we taped it down in place with more kapton. The sense coil has 13 turns.
Figure 8-27: Hand-wound sense coil.
The shielding helps to reduce any capacitive coupling from the actuator coil and
noise from other sources.
A key subtlety to the shield is that it must be open-
323
circuited: we tie one end to ground, but the other end we leave disconnected. The
reason for this is if we connected both ends of the shield to ground, we would have
a shorted turn around the pole face. This would result in eddy currents flowing in
the shorted turn and would affect our flux generation and flux measurement. This
configuration also provides self-shielding: since both the cable center and the shield see
the same potential induced by dB/dt, there will be no current in the cable center-toshield capacitance. Figure 8-28 shows the circuit diagram with actuator and amplifier
connections.
Shielded sense coil
Drive coil
Shedopen.-
Power
~~
1. -_.
-_wisted
pair--
77
amplifierd
DAC
- - - -.
-----
I-
----
Actuator
.--
---
I
AM502 diff amp
ADC
Shield grounded
-----------------------
-
Twiste pair AM502 diff amp Sense resistor (mounted on heat sink)
dSpace RTC
Figure 8-28: Circuit diagram of actuator setup.
8.4.10
Summary of Design Parameters
Table 8.1 lists some of the parameters of the finalized actuator design. These are also
the parameters that were used in the simulations of Chapter 4 through Chapter 6.
8.5
Reluctance Actuator Electrical Characteristics
In this section we calculate the resistive voltage, the inductive voltage, the speed
voltage, and maximum slew rate of the actuator to ensure that the amplifier is capable
of driving the actuator. We also predict the power dissipation.
324
Table 8.1: Reluctance actuator prototype design parameters.
Parameter
Core material
Pole face area
Actuator coil turns
Coil window area
Coil wire gauge
Sense coil turns
Value
49% Ni-49% Fe
726 mm 2
280
148 mm2
AWG #21
13
Resistive Voltage
8.5.1
We first calculate the coil resistance R. AWG #21 wire is listed in Section 20.3 of [48]
as having a resistance per length of r = 51 Q/km. The resistance R is then calculated
as
R = rNlavg,
(8.4)
where lavg is average turn length. The average turn length is 127 mm and the number
of turns is 280. This results in a predicted R = 1.5 Q. This is what we measured as
well. The resistive voltage,
VR,
is given as
(8.5)
VR= RI.
At a peak current (Ipk) of 5 A, we therefore expect the maximum VR to be about
7.5 V.
The peak dissipated power in the coil is I2R. We thus estimate a peak power of
37.5 W. Suppose we want to estimate how quickly the coil temperature will rise. As
a worst-case scenario, we can assume that all the heat is stored in the copper (i.e.,
the thermal resistance of the actuator is infinite). The temperature rise AT is given
by
AT
where
Q
Q
C
C'
(8.6)
is the heat energy dissipated by the current through a time At, C is the
thermal capacitance of the copper coil, and P is the average power dissipated. The
325
thermal capacitance is given by
C = mc"c,
(8.7)
where mc. is the copper mass and c is the specific heat capacity of copper.
The
specific heat capacity of copper is 385 J/(kg - K). The copper mass is given by
mcu = peV
=
d2
pcur-Nlavg,
4
(8.8)
where pu is the density of copper, V is the copper volume, d is the diameter of the
coil wire, and lag is the average turn length. The copper density pu is 8960 kg/m 3, d
for AWG #21 is 0.7239 mm, and lag is 127 mm. This results in m. = 0.131 kg. If we
assume a 50% duty cycle operating at peak power with a period of 0.1 seconds, then
P = 19W and for ten cycles (1 second duration), AT = 0.37K. This is a relatively
low temperature change, and indicates that we can run a number of cycles on the
actuator before having to worry about overheating. Suppose we want to maintain an
average coil temperature below 80'C. Then we set AT in (8.6) to 55 K (the change
in temperature from room temperature) and solve for At. This results in At
=
150 s
for a duty cycle of 50%. In other words, we can run the actuator at peak power with
a 50% duty cycle for two and a half minutes. This would give us over a thousand
cycles.
A more accurate estimate would need to include a thermal model of the whole
actuator, including the thermal resistance and the thermal capacitance of the housing
and the epoxy. It should also include the effect of temperature rise on the resistance.
This model was not computed herein. In any case, the actuator for a real device will
likely require the addition of liquid cooling, which is not undertaken here.
8.5.2
Inductive Voltage
From the analysis of Section 4.4.2, we can write the voltage
VA
from the flux linked
by the coil as
VA
= NA
A-
326
dt
(8.9)
We can replace B with (2.8) and write
poAN
2
d (I)
dt
g'
poA N 2
1 dI
2
2
V,\
gdt -
I dg
ci J*
(8.10)
The first term is the inductive voltage VL, which we write as
VL = L- =
dt
2g
(8.11)
dt'
where L is the inductance.
The foregoing analysis neglects the core reluctance. Thus, the analysis will not
be accurate near saturation.
For our prototype, if we assume an operating gap of
go = 500 pm, we can calculate the nominal inductance at this gap. We calculate a
value of 36 mH. A linearized circuit model of the actuator and amplifier at constant
gap is shown in Figure 8-29.
Actuator
A.V
vL
+
~
Rl
Amplifier
R
Figure 8-29: Circuit model of reluctance actuator.
The transfer function of the actuator circuit model is
I(s)
Vi(s)
AV,__(8_12)
_
Ls + R + R(
where R, is the sense resistor (0.1 Q in our setup) and A, the amplifier gain. The
corner frequency of the transfer function will occur at
327
f
=
(R+ R,)/(27rL). We deter-
mined the actual actuator inductance by measuring the frequency response from input
voltage (vi) to output current (I) of the actuator with a 500 pm gap. The measured
frequency response for several different current bias levels is shown in Figure 8-30.
The phase reaches 450 degrees at 5.5 Hz, indicating the corner frequency. Calculating the inductance from this value, we find L
=
46 mH, higher than predicted. One
possible reason for this is that our model does not take into account the leakage flux.
-0.15
-0.5
-1.0
1.5
100
--
A
A
A
A
2.0 A
10-2
10
10 3
102
104
-50
.......-
-100
. .
-150
10
102
frequency [Hz]
10 3
4
10
Figure 8-30: Frequency response of input amplifier voltage to reluctance actuator
current for different bias currents.
To determine the peak inductive voltage expected, we need to determine the peak
dI/dt. We can differentiate (2.8) and solve for dI/dt as
dI
2g dB
dt
poN dt
(8.13)
We can substitute (5.23) for dB/dt and write
dI
dt
dF
oAF dt
g
N
2
(8.14)
As shown in (8.14), the slew rate is inversely proportional to the square root of
the force. To avoid infinite slew rates, we must operate with some non-zero bias force
F0 . If the force profile is known, this equation permits us to calculate the expected
328
profile of the slew rate. For a typical scanning profile, the maximum value of dF/dt
is about 100 times greater than the numeric value of the maximum force, Fmax. If we
take Fmax for our prototype to be the maximum force simulated in Chapter 4, then
Fmax = 367 N, and our maximum dF/dt will be approximately 3.67 x 104 N/s. If we
use a 3rd-order position profile (i.e., finite jerk and infinite snap), then our maximum
dI/dt will occur at the beginning of the jerk phase when F = F.
We can write the
maximum dI/dt as
dI
g
dt max
N
2
dF
po AFo dt max'
where dF/dtmax is the maximum force slew rate.
Suppose that F = 1 N, then
dI/dtmax for our system will be 3.1 A/ms. Our maximum inductive voltage,
VL,max =
LdI/dtmax, will then be 140 V!
We can decrease the maximum inductive voltage by increasing F.
We can also
reduce VL by using a 4th-order position profile (finite snap), even while maintaining
equal or greater maximum dF/dt. The reason for this is that by using a 4th-order
profile, dF/dtmax no longer occurs at the minimum force equal to F.
The simulated
profile from Chapter 4 is derived from a 4th-order position profile. It has the same
dF/dtmax = 3.67 x 10 4 N/s that we calculated above. If we substitute F and dF/dt
from this profile into (8.14) and add a bias force of F = 1 N, we find a dI/dtmax
of 0.32 A/ms, a factor of nearly 10 lower than when using a 3rd-order profile.
We
calculate an associated VLmax = 14.5 V.
8.5.3
Speed Voltage
The second term in (8.10) is the speed voltage, v 8, which we write as
-
p0 APN 2 I dg
2g 2
dt
The motor constant Kmr is thus
Kmr
=
2g2
329
2.
(8.17)
We can double-check this motor constant by calculating &F/&I by differentiating
(2.14) with respect to I,
2
p
1 OAN I
aF
=
(8.18)
4g 2
If we note that A = 2AP, we see that OF/DI = K,.
The speed voltage is proportional
to I, so we expect the maximum v, to occur near the maximum current, Imax. We also
need to approximate the maximum speed, dg/dtmax. In Chapter 5 we approximated
dg/dtmax as 10 mm/s. Substituting this for dg/dt in (8.16) and 5 A for I, we estimate
a maximum v, of 3.6 V, which is small relative to the resistive and inductive terms.
8.5.4
Summary of Electrical Parameters
We summarize the lumped parameter electrical elements for the reluctance actuator in
Table 8.2. For the coil inductance and resistance, we include predicted and measured
values. We do not include values for the motor constant because it varies with current.
Table 8.2: Reluctance actuator prototype electrical parameters.
Parameter
Formula
Predicted Value
Measured Value
Resistance (R)
rNlavg
1.5 Q
1.5 Q
Inductance (L)
2g
36mH
46 mH
N/A
N/A
Motor Constant (Kmr)
poA$9
2
Table 8.3 summarizes the voltages, slew rate, and dissipated power expected in
the reluctance actuator.
We have assumed a bias force of FO = 1 N and the force
profile simulated in Chapter 4, which has a peak force of 367 N.
8.6
Voice Coil Actuator Electrical Parameters
Table 8.4 lists the electrical parameters of the voice coil actuator, as specified by BEIKimco. The measured inductance varies considerably from the specified inductance.
330
Table 8.3: Predicted peak electrical variables for reluctance actuator prototype.
Variable
Resistive voltage
Formula
Predicted Max Value
RI
7.5 V
LI
14.5 V
(VR)
Inductive voltage (VL)
Speed voltage (v,)
Slew rate (2)
dt
3.6V
N
1
Dissipated power (P)
O2AF
AFdt
I2R
0.32 A/ms
37W
Peak force (N)
367 N
This may be because the specified inductance was measured at 1 kHz, whereas we
calculated the inductance by measuring the frequency response of the actuator and
determining the corner frequency, as was done for the reluctance actuator.
Table 8.4: Voice coil actuator electrical parameters.
Parameter
Specified Value
Measured Value
Resistance (R)
Inductance (L)
Motor Constant (Kmvc)
2.6 Qt12.5%
2.9 mHt30%
35.14N/A 10%
2.9 Q
7.5 mH
30.7N/A
From these parameters, we can calculate the expected peak electrical variables.
For dI/dt, we write
dl
dt
1
dF
Kmvc dt
Since Kmvc is constant, the peak dI/dt will occur at peak dF/dt. The peak electrical
variables for the voice coil actuator are summarized in Table 8.5.
The 13 A listed as the peak current is slightly above the specified safe peak current
from BEI-Kimco, which is 12.7 A. For this reason, we will test the prototype actuator
at a maximum force level below 400 N. The dual channel Crown DC300A amplifier
331
Table 8.5: Predicted peak electrical variables for voice coil actuator.
Variable
Peak current (Ipk)
Resistive voltage
(VR)
Inductive voltage (VL)
Formula
Predicted Max Value
Kmvc
Fa.
13 A
RI
37.7 V
9.0 V
L
Speed voltage (v,)
-K2
0.3V
Slew rate ()dt
1 dF
Kmvc
dt
1.2 A/ms
12 R
490 W
Dissipated power (P)
we use to drive the actuators is capable of delivering a minimum of 155 W RMS
per channel. Since our predicted peak power dissipated for the voice coil actuator is
490 W, we should run the actuator at a duty cyle of 30% or less. Figure 8-31 shows
the VI limits for the Crown amplifier. The DC limits of the amplifier indicate that
it can source sufficient current to the voice coil.
The slew rate limit of the crown
amplifier is 8 A/ms, so we have sufficient margin for both the voice coil actuator and
reluctance actuator.
8.7
Summary
In this chapter we presented the design of a 1-DoF testbed for testing reluctance
actuator force accuracy.
The testbed incorporates an air bearing for the moving
stage in order closely to approximate a zero-stiffness motion stage. We use a highresolution encoder for accurate position sensing and control and high-stiffness load
cells for force measurement.
We presented a detailed design of the reluctance actuator prototype. We learned
that grinding the cut-core pole faces causes the laminations to 'mushroom' out despite
332
TYPICAL LIMITS OF VI OUTPUT
V
out
30
IFt
7,1
;3r
k
-0
40s-20
UNE FUSE BLOWS
(DC SINGLE CHAN.) BLO
AT 1OA DC IF BOTH
CHANNELS ARE DRIVEN
EQUALLY
0
-
-
00e 0 AC
X
WS
out
0
-
2
40
MID-FREQUENCY
6o
BURST
LIMIT
SHORT CIRCUIT CONTINUOUS LIMIT
' AREA OVER WHICH LIMITER (AC)
VARIES (SIGNAL DEPENDENT)
MAX. AC UMIT (k V OUT (SINE) AT
MAX.)
MAX. CONT. AC POWER
(NA = 2.75.n. I
HIGH FREQUENCY LIMIT
Figure 8-31: Crown amplifier VI limits. Figure taken from [30].
measures taken to prevent this. Moreover, it is not needed in order to ensure pole face
flatness. We therefore recommend not grinding the faces. We also determined that
the shield of the sense coil must be open-circuited to avoid eddy currents flowing in
the shield. We calculated predicted peak electrical variables for both the reluctance
actuator and voice coil actuator to ensure that our amplifier could drive the actuators.
We learned that we can reduce the slew rate and inductive voltage by using a
4 th_
order position profile rather than a 3rd-order position profile. In the next chapter, we
present experimental results from the reluctance actuator prototype using the 1-DoF
testbed.
333
334
Chapter 9
Loop-Widening Phenomenon
In this chapter, we document an investigation into unexpected loop-widening behavior
that the reluctance actuator prototype exhibited'at low frequencies.
In the course
of measuring the actuator outputs on the 1-DoF air bearing setup, we discovered
that the flux estimated from the sense coil exhibited a phase that lag relative to
the actuator current that increased with frequency.
This loop widening manifests
itself when plotting the flux against the actuator current for different frequencies: the
hysteresis loop widens with increasing frequency. While loop widening is expected
at higher frequencies, we discovered the phenomenon at low frequencies (< 10 Hz)
where eddy currents should not affect the hysteresis loop.
After discovering the
loop widening phenomenon on the air bearing setup, we embarked upon a series of
experiments in an effort to discover the root cause of the behavior.
This chapter is organized as follows.
We first present a series of experiments
performed on several actuator prototypes using both NiFe prototypes and CoFe prototypes. We then present experiments done on wound cut-cores. These were done in
order to isolate any effects from the cores themselves from any effects that might arise
from the actuator assembly process. Next we show the results of experiments that
were performed on ferromagnetic toroidal specimens. In this section we demonstrate
the difference in behavior between 49% NiFe and 50% NiFe. We then present a series
of experiments designed to uncover any problems in the instrumentation chain. The
outcome of our investigation was inconclusive and so we recommend further research
335
to uncover the root cause.
9.1
Actuator Experiments
To provide context, Figure 9-1 shows a representative series of B-I loops measured
on the reluctance actuator prototype on the 1-DoF air bearing setup. The zoomed-in
plot on the right clearly shows that the loops widen with increasing driving frequency.
B v.
1.5
I
B v.1
0.01
-0.5 Hz
-- 1 Hz
2 Hz
10.005
E
C
E
0.5
0
-5Hz
0
X
X
3 -0.5
-0.005
-1
-3
-2
-1.5
-1
L-0.011
1
current [A]
0
2
3
-0.1
-0.05
0
0.05
0.1
current [A]
Figure 9-1: LEFT: B-I data measured on reluctance actuator demonstrating loopwidening phenomenon; RIGHT: zoomed in near origin.
Because the relationship between actuator force and air gap flux density is not a
dynamic one (see (2.13)), we expected no phase lag between the measured force and
flux. However, when plotting the force measured by the load cells mounted to the
stator against the flux computed from the sense coil measurement, we found that the
force leads the flux. Figure 9-2 shows a measured B-F curve that was driven with a
10 Hz voltage sine wave. The red arrows show the direction of the loop made by the
B-F curve, indicating that the flux as measured by the sense coil lags the load cell
force measurement.
Furthermore, when we plotted the measured force against the measured current,
the loop width remained relatively constant with increasing frequency.
Figure 9-3
shows a series of measured F-I curves at different frequencies. These plots are taken
from the same data sets that were used in Figure 9-1. Any loop widening present
appears much reduced compared to the B-I curves shown in Figure 9-1. Moreover,
the direction of the F-I loop (counter-clockwise) indicates that the force lags the
336
............
F v. B
300
250200-
Direction
of loop
Z 150 -
100
-
.O
500
-50
1.5
1
0.5
0
flux density [T]
Figure 9-2: Measured B-F curve showing that the force leads the flux density.
current, as expected owing to hysteresis.
Fv. 1
F v.1I
155
300
250
---
0.5 Hz
-1
Hz
2 Hz
Hz
-5
2000 150-
8150-
0
0
100500
0
0.5
1
1.5
2
2.5
145
1.9
3
1.95
2
2.05
2.1
current [A]
current [A]
Figure 9-3: LEFT: F-I data measured on reluctance actuator; RIGHT: zoomed in
near 2A.
The problem that arises from loop widening is that it becomes much more challenging to generate accurate force control with the reluctance actuator. The purpose
of using a high-bandwidth flux loop to control the actuator is that it ideally allows for
accurate force control without having to resort to complex dynamic feedforward force
models of the actuator. If the measured flux exhibits phase lag that is not present in
the actual force, it becomes more difficult accurately to track a dynamically varying
force since the mapping between the measured flux and force is no longer a static one.
The fact that the measured force leads the measured flux and that the loop widening is not apparent in the F-I data indicated that the problem was an instrumentation
337
problem. To investigate the cause of the loop widening, we took a reluctance actuator
stator and clamped a target to it. This way we could isolate the actuator effects from
any other effects caused by the air bearing setup. This clamped setup is shown in
Figure 9-4.
Figure 9-4: Clamped actuator setup.
The instrumentation diagram of our clamped setup is shown in Figure 9-5. The
sense coil shield is tied to the cable shield on one end. On the other end, the sense
coil shield is disconnected and left open-circuited to avoid external eddy current loss
and for self-shielding from capacitance effects.
338
Shielded sense coil
Drive coil
arc
dpenpair
. .- . - -.. -- . - Tvwsted
..
- - --
DAC--
-
__Power
amplifier
AM502 diff amp
-.- - .. -
..
-.
ADC
- "Shield grounded
-
-
~TWistd pair
AM502 diff amp
-
-
,-----------------------
Actuator
Sense resistor (mounted on heat sink)
dSpace RTC
Figure 9-5: Instrumentation diagram for actuator experiments.
9.1.1
Gaussmeter Measurements
We used a Gaussmeter to measure the air gap flux independently.
We placed the
Gaussmeter probe in the center pole face air gap and taped it to the center pole face.
We placed delrin spacers in the left and right air gaps and then clamped the target
to the stator. The delrin spacers were slightly thicker than the Gaussmeter probe so
that the probe itself was not clamped. The total air gap was about 1.55 mm. The
Gaussmeter was an F.W. Bell 5180 model [69]. Figure 9-6 shows a series of B-I curves
measured with both the sense coil (left plots) and the Gaussmeter (right plots). The
Gaussmeter data is significantly noisier than the sense coil estimate; nevertheless,
loop widening is still evident from the Gaussmeter data. This suggests that the cause
of the loop widening is the actuator itself rather than the sense coil instrumentation
chain.
9.1.2
Actuator Simulations
We ran a series of actuator simulations to see if eddy currents could account for the
loop widening phenomenon. We used the model described in Section 6.6.3 for the
simulations with (5.38) used to model the excess eddy currents.
339
Equation (5.38)
Sense coil estimate
0.5
Gaussmeter measurement
0.5
E
0-
0-
-0.5
-2
-1
0
1
-0.5
2
0
1
2
0.005-005
0
-0.005
-0.01
-0.1
0
1 Hz
-2 Hz
5 Hz
-10 Hz
X
-
-1
0.01
0.01
E
-2
-0.05
0
current [A]
0.05
Hz
-
-0.005
-0
-
-0.01
H
-2
-1
0.1
-0.1
-0.05
0
current [A]
0.05
z
Hz
0H
0.1
Figure 9-6: LEFT: Measured B-I curves with B estimated from sense coil; RIGHT:
Measured B-I curves with B measured by Gaussmeter. Bottom plots are zoomed-in
graphs of the top plots.
describes the behavior of excess eddy currents in 50% NiFe. Our prototype uses 49%
NiFe. It is unknown whether 49% NiFe can be described by same model as 50% NiFe.
However, since (5.38) resulted in simulations with larger excess eddy currents than
the square root law of (5.27) that many other materials have been found to follow,
we used (5.38) since it should result in more loop widening than (5.27).
Figure 9-7 shows the simulation results for the model when driven with a 1 A
amplitude sine wave. There is some loop widening, but it is much less than what was
measured in the experimental results.
We applied a Fast Fourier Transform (FFT) to the current and flux density simulated data sets to gain more insight. At the fundamental frequency of each data
set, we computed the phase difference between the current and flux density FFT's.
Table 9.1 summarizes the results of this analysis. The function B(jw)/I(jw) refers to
the FFT from the input current to the flux density output. The phase lag increases
about 0.10 from 1 Hz to 20 Hz. In contrast, the measured data showed an increase
of ~0.8 from 1 to 20 Hz (see 'Center Coil' columns of Table 9.2). It therefore seems
340
B v. I
B v. I
0.02
0.5
E
E
2:
0.01
0
ci)
-2
I
-0.020.1
-0.1
'
-A
-1
-0.5
0
current [A]
0.5
F
1
-
-0.01
1 Hz
Hz
5Hz
Hz
20 Hz
-10
0
-0.05
0.05
0.1
current [A]
Figure 9-7: LEFT: Simulated B-I curves at different frequencies; RIGHT: zoomed in
near origin.
unlikely that eddy currents account for all of the loop widening in the prototype. If
we simulate the excess eddy currents with the square root dependency of (5.27), the
loop widening from eddy currents is even more negligible: the increase in phase lag
from 1 to 20 Hz is under 0.010.
Table 9.1: Phase between simulated actuator current and flux density
Frequency [Hz]
1
2
5
10
20
9.1.3
Z B(w [0]
-0.30
-0.33
-0.36
-0.39
-0.41
Multiple-pole Sense Coil Measurements
We added an independent sense coil wrapped around one of the outer pole faces
to compare with the center pole sense coil. We clamped the target to the stator
with 500 pm plastic shims. We acquired the data using a LabVIEW data acquisition system with its own differential front ends [64] instead of the dSPACE RTC
and AMD502 buffers (see Section 9.4.4 for more details on the LabVIEW acquisition
system). Figure 9-8 shows the B-I curves estimated from the center sense coil and
the outer sense coil. The bottom plots are zoomed in near the origin. While the flux
341
density estimated from the outer sense coil approximately reaches the same maximum flux density as center sense coil, the zoomed-in plots demonstrate that there is
some discrepancies between the loop shapes. For example, from the center coil flux
estimate, the 1 Hz loop the narrowest, while with the outer coil flux estimate, the 2
Hz loop is the narrowest.
Center pole face
0.5
Outer pole face
0.5
E
0
0-
-0.5
-1.5
-1
-0.5
0
0.5
1
-0.5
-1.5
1.5
0.02
E
0.01
-1
-0.5
0
0.5
1
1.5
0.02
-0.01-
0
-
0-1
-2
Hz
Hz
5 Hz
-0.01
-10Hz
---- 20 Hz
-2
-0.01
-
---
-0.02
-0.1
-0.05
0
current [A]
0.05
-io
-0.02
0.1
-0.1
-0.05
0
current [A]
0.05
1 Hz
Hz
5Hz
Hz
20 Hz
0.1
Figure 9-8: LEFT: Measured B-I curves estimated from center pole face sense coil;
RIGHT: Measured B-I curves estimated from outer pole face sense coil. Bottom
plots are zoomed-in graphs of the top plots.
We applied a Fast Fourier Transform (FFT) to the current and sense coil voltage
data sets to gain more insight. At the fundamental frequency of each data set, we
computed the phase difference between the current and sense coil voltage FFT's.
Table 9.2 summarizes the results of this analysis. The function V 8c(jw)/I(jw)
refers
to the FFT from the input current to the center sense coil voltage output. Likewise,
V,,(jw)/I(jw) refers to the FFT from the input current to the outer sense coil voltage
output. In these columns are listed the FFT phases of the fundamental frequency
component corresponding to the driving frequencies of the leftmost column.
The
functions B,,(jw)/I(jw) and B,(jw)/I(jw) are the respective center coil and outer
coil FFT's from current input to flux density. The fundamental component phases
342
were computed by subtracting 900 from the fundamental frequency component phases
of V c(jw)/I(jw)
and V 8 (jw)/I(jw).
Table 9.2: Phase between actuator current and sense coil voltage
Outer coil
Center coil
ffw)I(jw)
Frequency [Hz]
1
2
5
10
20
L 78'cCo) [0]
Bc(jw) 0
89.66
89.60
89.36
89.06
88.84
-0.34
-0.40
-0.64
-0.94
-1.16
IUWo)
1(jw)
Z[]
U Z(jO
89.84
89.96
90.25
90.65
91.02
B-UWJ) [0]
-0.16
-0.04
0.25
0.65
1.02
From the B,,(jw)/I(jw) column, we see that the center coil flux lags the current
and that this phase lag increases with frequency, confirming the conclusions drawn
from the graphical analysis. The outer coil flux, however, tells a different story. Here,
the flux lags the current at frequencies of 1 Hz and 2 Hz, but leads the current at 5 Hz
and higher. In contrast to the center pole flux, the phase lag from the current to the
flux decreases with frequency so that it actually becomes a phase lead at frequencies
of 5 Hz and above. The center coil and outer coil thus show opposite loop-widening
behavior. When we inspect the right-hand graph of Figure 9-8 more closely, we see
that B-I loops corresponding to frequencies of 5 Hz and greater travel in a clockwise
direction rather than in a counter-clockwise direction, graphically verifying that the
flux leads the current at these frequencies.
9.1.4
CoFe Lab Prototype Actuator
We tested a CoFe actuator prototype designed and built by Roberto Melendez [62].
Figure 9-9 is a photograph of the actuator stator in a test fixture that Roberto used
for his thesis. The test fixture was not used for the loop-widening experiment, but
rather the actuator was clamped in a similar fashion to the setup shown in Figure 9-4.
The actuator is a two-pole actuator rather than a three-pole actuator. The target
used was a NiFe target because CoFe targets were not available at the time of testing.
343
We wound a sense coil by hand around one of the actuator pole faces.
Figure 9-9: CoFe actuator prototype stator.
Figure 9-10 show the measured A-I curves at a gap of 500 pm, where A is the
total flux linked by the sense coil. Loop widening appears to be less pronounced for
frequencies 10 Hz and below, although some loop widening is apparent for the 20 Hz
and 50 Hz curves.
X 10-3
6 v.I
6
v.I
-
0.06
0.04-
-
-0.02
0)
0Y)
0
--
-1
H
0 -00
-
x -0.02
-0.04
-20
--
-0.06
-3
-2
-1
0
1
2
5
-0.2
3
current [A]
-0.1
0
0.1
Hz
Hz
50 Hz
0.2
current [A]
Figure 9-10: LEFT: Measured A-I curves from CoFe lab prototype actuator; RIGHT:
Zoomed in near origin.
Table 9.3 summarizes the FFT analysis of the data shown in Figure 9-10. The
CoFe actuator shows less loop widening than the NiFe actuator.
9.1.5
CoFe ASML Prototype Actuator
We also tested a CoFe prototype actuator that we designed at ASML. This actuator
is a two-pole actuator. It was designed for the full force capability required for the
344
Table 9.3: Phase between CoFe lab prototype actuator current and sense coil voltage
Frequency [Hz]
(o
-0.28
-0.28
-0.34
-0.39
-0.51
-0.64
89.72
89.72
89.66
89.61
89.49
89.36
1
2
5
10
20
50
1
[2(
I]
0
L
reticle short-stroke stage and is capable of generating over 1000 N of force. For this
actuator, we also had a CoFe target with which to test it. We tested it at both 500 Jrm
gap and a 1 mm gaps. Figure 9-11 shows series of measured B-I curves at both of
these gaps. The curves appear to show less loop widening than the NiFe actuator for
frequencies 10 Hz and below, although some loop widening is apparent for the 20 Hz
curve.
1 mm gap
0.5 mm gap
2-
E
1-
E
-7
4) 0x
"a
-1
-1:
-2
2
-1
0
-2
2
1
01
E
....
...
...
0
2
-1
0.05
0.05
(D
25
0
Hz
2 Hz
5Hz
-10 Hz
-20 Hz
-1
-0.1
-0 .1
--
-2
--
2 Hz
Hz
5 Hz
-0 Hz
20 Hz
'
-0.05
0
-
x
'a
x0
-0.05
0
0.05
-0.1
0.1
-0.05
0
0.05
0.1
current [A]
current [A]
Figure 9-11: LEFT: Measured B-I curves from ASML CoFe prototype actuator at
500 prn gap; RIGHT: Measured B-I curves from ASML CoFe prototype actuator at
1 mm gap. Bottom plots are zoomed-in graphs of the top plots.
The FFT analysis results for the ASML prototype are presented in Table 9.4.
345
The results indicate the presence of loop widening; however, this loop widening is
considerably less than the amount present in the NiFe actuator. For example, between
1 Hz and 10 Hz, there is an additional 0.140 of phase lag in the 500 pm gap case,
whereas for the NiFe actuator, we see an additional 0.30 of phase lag (see 'Center
coil' columnn in Table 9.2).
Table 9.4: Phase between actuator current and sense coil voltage for ASML CoFe
prototype actuator.
Frequency [Hz]
1
2
5
10
20
9.2
0.5mm gap
BZ O
wVo(J) [0] Z Bc(jw) [0]
L ~cJw) []
I(j)
89.65
89.63
89.60
89.51
89.42
OW)
-0.35
-0.37
-0.40
-0.49
-0.58
1mm gap
I(jW)
89.75
89.72
89.66
89.62
89.46
I(jO)
-0.25
-0.28
-0.34
-0.38
-0.54
Cut-core Experiments
In an effort to isolate potential causes of the loop widening, we measured the fluxcurrent relationship of cut-cores that were not potted and incorporated into a stator
housing. We experimented on both a 49% NiFe cut-core and a CoFe cut-core.
9.2.1
49% NiFe Cut-core
Figure 9-12 shows the experimental setup for testing a 49% NiFe core. This core
is identical to the ones used in our actuator prototype. Two coils are hand-wound
around the full length of the cut-core in a bifilar fashion. The cut core is then clamped
to an assembled target with 500 pm plastic shim between the core and the target. The
primary coil is connected to the sense resistor on one end. The secondary coil is used
as the sense coil. The inductance of the primary with a 500 pm gap was computed as
9.3 mH. The primary coil resistance was 1.5 Q. These values correspond to a corner
frequency between drive voltage and current of 26 Hz.
346
49% NiFe core with
hand-wound primary
and secondary coil
500 pm shim
Target
Figure 9-12: Experimental setup for measuring B-I characteristics of single 49% NiFe
cut-core.
Figure 9-13 shows the A-I loops measured on the cut-core when driven at different
frequencies. The measured loops appear to show much less loop widening than the
NiFe prototype actuator exhibited. The loops corresponding to frequencies 1-10 Hz
are on top of one another. The 50 Hz loop is the first one to show noticeable loop
widening.
X 10*3
'
4
6 v.1
5
-
6
6 v.1
X10-
4
0
>
-1
-2
-2
-10
-20
-50
-4
-62
-1
0
1
-51
-0.1
2
-0.05
0
0.05
Hz
Hz
5 Hz
Hz
Hz
Hz
0.1
current [A]
current [A]
Figure 9-13: LEFT: Measured A-I curves from 49% NiFe cut-core; RIGHT: Zoomed
in near origin.
347
We applied an FFT analysis to this data. Table 9.5 summarizes the results of this
analysis. The function V(jw)/I(jw) refers to the FFT from the drive coil current
to the secondary coil voltage output. The functions A(jw)/I(jw) is the FFT from
drive coil current to secondary coil flux linkage. The FFT analysis confirms that for
frequencies between 1-5 Hz, there is negligible loop widening. The 10 Hz loop shows
an increase in phase lag of ~0.1
-0. 1
0
of phase lag.
and the 20 Hz loop likewise shows an additional
These are smaller changes than the corresponding phase lag
changes in the actuator, where the phase lag increased by 0.3
by ~0.2
from 5 to 10 Hz and
from 10 to 20 Hz (see Table 9.2). On the other hand, the phase lag from
the current to flux linkage in the cut-core is greater at low frequencies than that
manifested by the actuator.
For example, at 1 Hz, the phase lag measured in the
actuator center pole is 0.34' while the phase lag measured in the cut-core is 0.720.
Table 9.5: Phase between 49% NiFe cut-core current and sense coil voltage
Frequency [Hz]
1
2
5
10
20
50
9.2.2
Z '(w)[0]
89.28
89.29
89.27
89.15
89.03
88.88
j?
0]
-0.72
-0.71
-0.73
-0.85
-0.97
-1.12
CoFe Cut-core
Figure 9-14 is a photograph of the CoFe cut-core that we tested. The details of this
cut-core can be found in [62]. A machine-wound coil assembled by Roberto Melendez
was used as the drive coil. A hand-wound coil was used as the secondary coil. Each
coil was placed around separate poles. The setup for testing was otherwise identical
to that shown in Figure 9-12. A 49% NiFe assembled target was used instead of a
CoFe target because the CoFe I-bars had not arrived at the time of testing.
Figure 9-15 shows the measured A-I curves at different frequencies. The CoFe core
348
Figure 9-14: Photograph of CoFe cut-core with machine-wound primary coil on left
and hand-wound secondary sense coil on right.
shows very little loop widening for the frequencies tested. An FFT analysis confirms
that the phase lag variations at the different frequencies remain within 0.050 of one
another.
3
6 v. I
0
0.03
6 v. I
x10-
0.02
a)
0)
CU
d)
0.01
0)0
0
-0.01
-1
--
-1
-0.02
-0.03
-10
3
-2
-1
0
current [A]
1
2
-0.2
3
-0.1
0
0.1
Hz
2 Hz
5 Hz
Hz
-
0)
0.2
current [A]
Figure 9-15: LEFT: Measured A-I curves from CoFe cut-core; RIGHT: Zoomed in
near origin.
9.3
Material B-H Experiments
In an earlier set of experiments, we tested the magnetic properties of several ferromagnetic materials. We discovered loop widening at low frequencies on a 50% NiFe
specimen but not on a 49% NiFe specimen.
349
9.3.1
50% NiFe
Figure 9-16 is a photograph of a 50% NiFe toroidal specimen with bifilar windings.
The lamination thickness is 0.002" (~50 1m). The inner diameter is 2", the outer
diameter is 2.5", and the height is 0.5". The inner winding used as the drive coil has
150 turns and the outer coil used as the sense coil has 120 turns.
Figure 9-16: Photograph of 50% NiFe toroid with bifilar windings.
Figure 9-17 shows a series of B-H curves measured on the specimen at different
frequencies. The primary coil was driven with a voltage sine wave. The magnetic field
H was measured with a sense resistor connected to the drive coil and the magnetic flux
density B was calculated in post-processing from numeric integration of the secondary
coil voltage. The B-H curves show clear loop widening, even at low frequencies.
We discovered that the reason for this loop widening is because of the steep transition in the B-H loop near the coercive field.
This results in B having a higher
frequency content than the driving signal. Figure 9-18 demonstrates the sense coil
voltage signal v, when the primary coil is driven with a 1 Hz voltage sine wave. Note
the high-frequency spikes in the v, signal. These correspond to traversing the high
dB/dH portions of the B-H loop.
We can also view this phenomenon from a mathematical perspective. Using the
chain rule, we can express dB/dt as
dB
dBdH
dt
dH dt
350
B v.H
1.5
1
i~. 0.5 F
C
a)
*0
0
x
i2F Hz
Hz
-0.51
-1
-20
--
4 F Hz
10 Hz
Hz
40 Hz
I
0
magnetic field [A/m]
-1.5'
-50
50
Figure 9-17: Measured B-H curves on 50% NiFe toroid specimen.
B v. H
0.3
.
1 5
15--
E 0.
>
0-
1 -40
-V
0.1
Large dB/dt~
0
-0.1
-0. 5
5
0.2
-20
J
F
0
magnetic field [A/m]
-0.2
-0.1
Large dB/dH
20
40
0
0.5
1
1.5
2
time [sec]
Figure 9-18: LEFT: Measured B-H loop on 50% NiFe toroid driven at 1 Hz; RIGHT:
Measured sense coil voltage v,.
From Ampere's Law, the magnetic field rate of change dH/dt is proportional to the
current rate of change dI/dt. Because of the nonlinear nature of dB/dH, even if
the current input is driven at a low frequency, the output dB/dt will have higher
frequency content than the input signal. Exciting this high-frequency content can
result in eddy currents reducing the flux measured by the sense coil. This is what we
see in the B-H measurements on the 50% NiFe sample.
9.3.2
49% NiFe
We also measured the B-H characteristics of 49% NiFe.
Some of the differences
between 50% NiFe and 49% NiFe were previously discussed in Section 7.5.6. A pho-
351
tograph of the specimen is shown again in Figure 9-19. The specimen is wound with
two coils in a bifilar fashion. The inner coil, used as the drive coil, has 116 turns and
the outer coil, used as the sense coil, has 108 turns. The specimen is laminated with
0.001" (-25 prm) thick laminations.
10.8 cm
Figure 9-19: 49% NiFe sample used to measure B-H curves.
Figure 9-20 shows the measured B-H curves at three different frequencies of the
49% NiFe specimen. The primary coil was driven with voltage sine waves. The loops
show negligible loop widening.
Note that the B-H loop is much less steep at the
transition region from negative saturation to positive saturation than the 50% NiFe
B-H loop. This may account for why we see loop widening at low frequences in the
50% NiFe but not in the 49% NiFe.
We also measured two 49% NiFe cut-cores positioned face-to-face to approximate
a toroidal shape. These cut-cores have a lamination thickness of 0.004" (~100 pm),
identical to the lamination thickness in the NiFe actuator prototype. Testing these
cut-cores helps us to determine whether the absence of loop widening at low frequencies for the 49% NiFe specimen shown in Figure 9-19 is due to the thinner laminations
or to the relatively low dB/dH in the transition region from negative B to positive
B. The cut-cores are smaller than the ones we used in the prototype actuators. The
pole face dimensions of a single pole are 0.25" xO.5", compared to 0.375" xO.75" for
352
B v.H
1.5
1
PT 0.50-
/
'0
-0.5
/
-
1Hz
2 Hz
4 Hz
-1.5
-
-1
-300
-200
100
0
-100
magnetic field [A/m]
200
300
Figure 9-20: Measured B-H curves on 49% NiFe specimen.
the cut-core used for the prototype actuator. Figure 9-21 shows a photograph of one
of the cut-cores and the setup used for clamping the two cut-cores together.
The
drive coil is wound with 25 turns on one cut-core, and the sense coil is wound with
85 turns on the other cut-core.
..........
Primary o
condary
Nonma
spacer
2.5
mr
Figure 9-21: LEFT: 49% NiFe cut-core used for testing B-H properties; RIGHT:
Experimental setup for testing B-H properties.
Figure 9-22 shows the measured B-H loops for three different frequencies on the
clamped cut-core setup. We drove the primary coil with voltage sine waves. We see
negligible loop widening at low frequencies, giving us further evidence that the lower
dB/dH near the coercive field is what distinguishes the 49% NiFe behavior from the
353
:netic
50% NiFe behavior.
B v. H
0.2
0.1
E7
0
-0.1
-0.2'
-30
-- 1 Hz
-- -2 Hz
4Hz
-20
-10
0
10
magnetic field [A/m]
20
30
Figure 9-22: Measured B-H curves on 49% NiFe cut-cores.
9.4
Instrumentation
In this section, we present our investigation into the instrumentation chain to determine if this was causing the loop-widening behavior. We document experiments with
an air core toroid. We demonstrate on the air core that the current sense resistor
value affects the amount of loop widening. We then investigate the affect of the differential amplifier gain with the reluctance actuator prototype and discover that the
amplifier gain also affects the amount of loop widening. These two results indicate
that instrumentation problems are at least a partial cause of the loop widening. We
then demonstrate results using a LabVIEW data acquisition system with its own
differential front-end and our own instrumentation front-end built with AD620 differential amplifiers. Both of these options improve the results but do not completely
eliminate the loop widening.
9.4.1
Air Core Test Setup
To debug the instrumentation chain, Professor Jeffrey Lang suggested that we test
an air core. This way, we eliminate ferromagnetic effects from the test. The air core
354
should exhibit no phase lag between the drive current and the flux estimated from
the sense coil. If phase lag persists, this indicates a problem with the instrumentation
chain.
Figure 9-23 is a photograph of the air core with primary and secondary coils. A
roll of electrical tape was used as the 'air core'. The coils were wound in a bifilar
fashion, with the inner coil consisting of 150 turns and the outer coil consisting of 100
turns. The instrumentation chain was identical to that shown in Figure 9-5 except
for the air core replacing the actuator. Additionally, the sense coil is not shielded. A
photograph of the testing setup is shown in Figure 9-24.
Figure 9-23: Photograph of 'air core' with primary and secondary coils.
9.4.2
Current Sense Resistor
One of the hypothesized causes of the loop widening was sense resistor heating: as
the resistor heats up, the resistance increases and gives an erroneous current measurement. Because of the slow time constant associated with the temperature rise, this
could result in a phase lag between the estimated flux and the measured current. To
test this hypothesis, we switched the 0.1 Q sense resistor with a 0.01 Q sense resistor
355
Figure 9-24: Experimental setup for testing air core.
with identical volume. The smaller sense resistor will nominally dissipate ten times
less power for the same current, which will limit its temperature rise. Figure 9-25
compares the measured flux-current loops on the air core using the 0.01 Q resistor
with the measured flux-current loops using the 0.1 Q resistor. In an effort to reduce
noise, the plotted variables are cyclic averages of 10 to 20 cycles at each frequency.
We also set the low-pass filters on the AM502 amplifier channels to a 1 kHz cutoff
frequency. We set the gain on the AM502 sense coil channel to as high as possible
while still avoiding saturation.
Contrary to expectations, the data obtained with the 0.1 Omega sense resistor
356
R =0.01 Q
0.5
0.5-
0
-
-
S0-
0.5-0-
X -0.5 -.P-
12-1
X106
X
x10-6
1 Hz
2 Hz
--
-1
-2
--
00 -
0.5
.z
5
-0.5
-10
10 Hz
20 Hz
50 Hz
)
---
0
--
Hze
10
x-0.5 -
.- 0.5 -
-0.05
5
0
-5
5
0
-5
0)
R, = 0.1
X 10.4
1
1 X 10
0.05
0
-0.05
Hz
Hz
5 Hz
Hz
20 Hz
50 Hz
100 Hz
0
00
0
0.05
current [A]
current [A]
Figure 9-25: LEFT: Measured A-I curves on air core with 0.01Q sense resistor;
RIGHT: Measured A-I curves on air core with 0.1Q sense resistor. Bottom plots
are zoomed-in graphs of the top plots.
showed less loop widening than the data obtained with the 0.01 Omega sense resistor.
Note that both data sets show much less loop widening than the actuator did: the
bottom plots are significantly zoomed in order to show the loop widening that does
exist.
Applying an FFT analysis to the data sets generated the phase lag results
summarized in Table 9.6.
We see that the phase difference between current and
estimated flux linkage remains within 0.10 for frequencies up to 100 Hz when using a
0.1 Omega sense resistor. We hypothesized that the reason the smaller sense resistor
fares worse is because the voltage drop across the sense resistor is smaller relative to
the voltage drop in the power return line from the low end of the sense resistor to
power ground. As a result, the common-mode voltage across the resistor is higher
relative to the differential voltage.
Since the AMD502 instrumentation amplifiers
have finite CMRR, this could result in phase lag.
At a frequency of 100 Hz, we suspected that mismatches between the 1 kHz filters on the differential amplifier channels may be causing some of the small phase
difference between the current and flux. To test this, we switched the channels so
357
Table 9.6: Phase between air core primary coil current and secondary sense coil
voltage with different current sense resistors
Frequency [Hz]
R, = 0.1Q
R, = 0.01 Q
L XUw)
Zv.jw)
1
2
5
10
20
50
100
I_ ) _0_
90.002
89.992
89.927
89.877
89.777
89.589
89.445
[0
I(jW)
0.002
-0.008
-0.073
-0.123
-0.223
-0.411
-0.555
/Uw)
I(jW)
o
IUW)
0.005
0.005
-0.002
0.009
0.022
0.047
0.088
AUw)
[
90.005
90.005
89.998
90.009
90.022
90.047
90.088
that the channel we had connected to the sense coil was now connected to the sense
resistor and vice-versa. Figure 9-26 shows the measured A-I loops for the two different configurations when driven at 100 Hz. The plots are zoomed in near the origin.
The sense resistor value is 0.01 Q. We see that in the configuration with the differential amplifier channels switched, the phase difference between the current and flux
reverses, so that the loop now travels in a counter-clockwise direction rather than a
clockwise direction. This suggests that the filter mismatch between the two channels
does indeed account for some of the phase difference between the current and flux.
x 10-6 Differential Amplifier Channels Switched
Original Configuration
x 10-6
Direction
0
-
0--o
Direction
x
-5
-0.2
of loop
-
x
-0.1
0
current [A]
0.1
-0.2
0.2
-0.1
0
current [A]
0.1
0.2
Figure 9-26: LEFT: Measured A-I loop on air core; RIGHT: Measured A-I loop on
air core with AM502 differential amplifier channels switched.
We increased the sense resistor value to 0.2 Q to see if we could further improve
the results. We did not see any change from the 0.1 Q case. We likewise tried a 0.2 Q
resistor with the reluctance actuator, but saw no change.
358
9.4.3
AM502 Differential Amplifiers
The AM502 instrumentation amplifiers have finite CMRR. This CMRR increases with
amplifier gain. For example, the AM502 specifications list the CMRR as 100 dB for
gains greater than or equal to 100, but 50 dB for gains less than 100 [81]. With the
air core tests, we were able to set the amplifier gain on the sense coil channel to high
values because the flux levels were so low, and thus we were able to take advantage of
the high-gain CMRR. Up to this point, when testing the reluctance actuator, we had
used a gain of ten on the sense coil differential amplifier channel to avoid saturation.
We decided to investigate the effect that amplifier gain had on loop widening.
Table 9.7 compares the phase difference between B and I for the NiFe reluctance
actuator prototype when the sense coil differential amplifier channel gain is set to
different gains. The left columns show the results using a high gain on the differential
amplifier channel: we selected the highest gain possible while still avoiding saturation.
This is why the gain is lowered as the frequency is increases: dB/dt and therefore
v, increases with increasing frequency, so the amplifier gain must be decreased to
prevent saturation. The right columns show the results using the original differential
amplifier gain setting of ten. The table shows that high gain settings result in lower
phase lags than the low gain setting. For example, a gain of 500 for a 1 Hz signal
leads to a phase lag of roughly half that measured with a gain of ten.
Table 9.7: Comparison of phase between actuator current and sense coil voltage for
different sense coil differential amplifier gain settings.
Low DA gain
High DA gain
Frequency
DA
[Hz]
gain
500
200
100
50
20
1
2
5
10
20
DA
Z V.U')[0]
89.64
89.56
89.34
89.02
88.73
zB() )[o]
-0.36
-0.44
-0.66
-0.98
-1.27
gain
10
10
10
10
10
1
. jw) [0]
89.26
89.40
89.28
89.00
88.73
z
?)
[01
-0.74
-0.60
-0.72
-1.00
-1.27
These results motivated us to improve our instrumentation chain. We considered
359
two options. The first option was to use a LabVIEW high-resolution data acquisition
system in place of the dSPACE DAQ and AM502 differential amplifiers. The dSPACE
1103 board that we had been using has 16-bit A/D boards with a mininum of 83 dB
of signal-to-noise ratio. In contrast, the LabVIEW A/D has 18-bit resolution and its
own differential front-end with a CMRR over 100 dB. The second option is to build
our own differential front end with high CMRR and use this instead of the AM502
channels. We investigated both options.
9.4.4
LabVIEW Data Acquisition System
We first tested the LabVIEW Data Acquisition System (DAQ) with the air core to
ensure that the instrumentation chain was working properly.
We obtained similar
results to those presented in Section 9.4.1 with the 0.1 Q sense resistor, i.e., showing
negligible phase lag between the current and flux.
When testing the reluctance actuator, we obtained similar results to those obtained using the AM502 high-gain setting. Table 9.8 summarizes the phase results
using LabVIEW. Loop widening at low frequencies is still present.
Table 9.8: Phase between reluctance actuator current and sense coil voltage using
LabVIEW DAQ
Frequency [Hz]
z;(j;) [0]
1
2
5
10
20
9.4.5
89.65
89.58
89.35
89.06
88.83
1
.W)
[0]
-0.35
-0.42
-0.65
-0.94
-1.17
AD620 Instrumentation Amplifier Front-End
We breadboarded two AD620 IC's to use as our differential amplifier front-end to our
data acquisition system. The AD620 is an instrumentation amplifier IC from Analog
Devices [6] with adjustable gain. The gain is set with a single resistor. As with the
360
AM502 amplifiers, the CMRR degrades as the gain is decreased. The typical CMRR
listed on the data sheet is 90 dB, 110 dB, and 130 dB for gains of for gains of 1, 10,
and 100, respectively. Nevertheless, these CMRR values are still significantly higher
than the AM502 CMRR at comparable gains. Figure 9-27 is a photograph of the
breadboarded circuit.
One AD620 chip is used to measure the voltage across the
sense resistor and the other is used to measure the sense coil voltage.
AD620
instrumentation amps
Figure 9-27: Experimental setup for testing air core.
As with the LabVIEW differential front-end, we first tested the AD620 front-end
with the air core. We achieved negligible phase lag between the measured current
and flux with the air core. With the reluctance actuator, we obtained similar results
to the LabVIEW results. Table 9.9 summarizes the phase results with the AD620
front-end. A gain of 10 was used for the current measurement AD620 and a gain of
100 was used for the sense coil AD620.
We also found that the gain on the AD620 sense coil channel affected the amount
of phase lag. For example, when we used a gain of 10, there was more phase lag from
the measured current to the measured flux than when we used a gain of 100. Thus,
the AD620 instrumentation amplifiers showed similar behavior to that of the AM502
amplifiers.
361
Table 9.9: Phase between reluctance actuator current and sense coil voltage using
AD620 front-end.
Frequency [Hz]
1
2
5
10
9.5
ZV
[0]
89.67
89.55
89.32
88.97
Zi
[Z]
-0.33
-0.45
-0.68
-1.03
Summary
We have documented a series of experiments undertaken to uncover the root cause of
the loop widening phenomenon at low frequencies with the prototype reluctance actuator. Actuator simulations suggest that the amount of loop widening is too large to
be explainable by eddy currents alone. This conclusion was confirmed by testing NiFe
cut-cores and measuring the B-H properties of 49% NiFe, which showed negligible
loop widening at low frequencies. Changing the gain on the differential amplifier and
changing the sense resistor value both resulted in different results. This indicated that
the instrumentation chain was part of the problem. To debug the instrumentation
chain, we tested an air core. We were able to demonstrate negligible loop widening
on the air core both with the AMD502 buffers using high gain settings and with the
Labview high-resolution DAQ. However, the problem reappeared when testing the
actuator with the Labview high-resolution DAQ, indicating that the problem is with
the actuator itself.
On the other hand, the Gaussmeter measurements showed similar loop-widening
behavior to that manifested by the sense coil measurements. This suggests that it is
not a measurement problem but that the actuator flux is being affected. However,
in that case, we would expect the measured force to show similar loop-widening
behavior when plotted against the current, but we do not. Moreover, the sense coil
measurements from the outer pole face showing phase lead with increasing frequency
are inexplicable with this hypothesis.
We do not at present have a hypothesis that accounts for the apparent contra-
362
dictions among the different tests and simulations. Time did not permit for a more
thorough investigation to uncover the root cause of the loop widening.
One crucial point that was driven home during this investigation is the importance of properly managing the electrical wires and connections, especially on the
instrumentation side. Appropriate grounding and shielding can solve a host of problems and is necessary for achieving high precision control of a reluctance actuator.
Moreover, as was demonstrated from the variation resulting from the choice of current
sense resistor and differential amplifier gain, we learned that careful of selection of
sensors and buffer amplifiers is critical to achieving high performance.
Another important insight gained during testing was the effect of the nonlinear
B-H curve on loop widening in 50% NiFe.
Because of the steep transition in the
B-H loop near the coercive field, the flux density can have high-frequency content
even when driven with a low-frequency magnetic field. This generates eddy currents
that cause loop widening.
Thus, we recommend that 50% NiFe not be used for a
reluctance actuator in a lithography machine.
In the next chapter, we present experimental results from the reluctance actuator
prototype using the 1-DoF testbed. We demonstrate the accuracy of the SHM, and
we present results of a flux feedback controller that combines the SHM and sense
coil estimates. We discuss the effects of the loop widening phenomenon on the force
tracking experimental results using the flux feedback controller.
363
364
Chapter 10
One-DoF Testbed Experimental
Results
In this chapter, we present experimental results for the reluctance actuator on the
1-DoF testbed. We first present the overall controller structure of the testbed and
then present controller designs for the voice coil current controller and the air bearing position controller. We then demonstrate calibration results for the reluctance
actuator from the 1-DoF testbed. We map current and gap to flux and map flux and
gap to force.
We implement the SHM observer from Section 6.6 to provide a low-frequency
estimate of the actuator flux. We compare the accuracy of the estimated output with
the sense coil measurement for different inputs. We also demonstrate the accuracy of
the hysteresis model when a gap disturbance is applied to the air bearing slide using
the voice coil actuator.
We then incorporate this SHM observer into a flux estimator. For the flux estimator, we employ a novel technique for estimating the flux, whereby we combine
the sense coil measurement with the SHM observer estimate using a complementary
filter pair. This results in a flux estimate for frequencies from DC to several kHz that
can be used as the feedback variable for the reluctance flux controller. This hybrid
flux estimator is incorporated into a flux feedback controller. We use the calibration
results to invert the desired force into a desired flux density that is used for the flux
365
loop command signal. We experimentally demonstrate the small-signal force tracking
ability and the stiffness of the actuator with flux control. Finally, we demonstrate the
force accuracy of the reluctance actuator with flux control for a force pulse similar
to that used on a scanning lithography machine. We repeat this experiment in the
presence of a gap disturbance.
10.1
Controller Structure
Figure 10-1 illustrates the overall controller structure of the testbed.
The linear
encoder signal xm is used as the position feedback variable. The position controller
(dotted blue box) tracks the desired reference position x,. The voice coil actuator is
used as the position feedback actuator and applies a force F,, to the air bearing slide.
Within the position loop, a inner current control loop (dotted green box) is used
to achieve high actuator bandwidth for the voice coil. A flux control loop (dotted
red box) is used to provide accurate force control for the reluctance actuator. An
inverse actuator calibration B(F, g) transforms the desired force Fd into a desired flux
density Bd. This inverse calibration uses xm as an input to account for the actuator
dependency on operating gap. The flux estimation scheme is used to generate a flux
density estimate
E for
the flux feedback variable. The reluctance actuator applies a
force F, to the air bearing stage. The desired force is also multiplied by -1/kmc
to
generate a feedforward current IFF to the voice coil controller. The purpose of this
is to help counteract the reluctance actuator force and achieve higher performance in
maintaining the desired operating gap. The parameter kmc is the estimated motor
constant of the voice coil actuator.
10.1.1
Voice Coil Current Controller
The measured frequency response of the uncompensated voice coil is shown in Figure 10-2. The input is the voltage to the amplifier and the output is the measured
current across the sense resistor. The measurement chain thus includes the amplifier
dynamics.
The sense resistor voltage was sent to an AM502 differential amplifier
366
Flux feedback
loop
-
CB
- B(Fd ,g)
F(B,g)
Actuator
Flux
controller r
Flux
Estimation
1FF
CF
C
Position
controller
w- F
x+
v
Enire
Voice coil current
controller
Current
feedback loop
-----------------------------------------------------------------
Position
feedback loop
Figure 10-1: Controller configuration used for 1-DoF testbed.
sent
channel with a low pass filter with cutoff frequency of 10 kHz. This signal was
to a dSPACE A/D channel in order to capture the phase delay from sampling. The
signal was sampled at 40 kHz.
We designed our voice coil controller to have a crossover frequency
The plant has a gain of 0.057 A/V and a phase of -112'
f,
of 5 kHz.
at 5 kHz. Our controller is
a simple PI controller with the zero frequency set at a factor of five below
fc.
The
controller form is
Ci(s)
We set K,
=
=
K1
T1s +
1
(10.1)
.
TIS
17.2 V/(A -s) and Tr = 1.59 x 10-4s.
The predicted and measured
frequency responses of the voice coil loop transmission are shown in Figure 10-3. The
crossover frequency of the measured loop transmission is -6.3 kHz, slightly higher
than predicted.
The reason for this may be that the input voltage amplitude is
different from that used to measure the plant frequency response from which the
predicted loop transmission was derived. The phase margin is ~45'.
367
I(s)IV(s)
100
10-2
1 00
10
3
102
10
10 2
10 3
10
4
0
a) -100
_0
a)
U.
.- -200
0L
-300
100
101
104
frequency [Hz]
Figure 10-2: Measured frequency response for uncompensated voice coil actuator.
Voice coil loop transmission
-
105
predicted
measured
0)
E
1
10-5
1
I0
10
102
103
104
102
103
10 4
-50
-100
0)
a)
:C -150
CO
-200
-250
-300
100
101
frequency [Hz]
Figure 10-3: Predicted and measured frequency responses for voice coil loop transmission.
368
Position Controller
10.1.2
The measured frequency response of the air bearing slide is shown in Figure 10-4. The
slide was driven with the voice coil actuator. The input for the measured frequency
response is the voice coil current I,, and the output is the encoder measurement
xm. Since the response was measured open-loop, we placed soft compression springs
between the air bearing stator and the reluctance actuator target to ensure that the
slide did not drift and cause the reluctance actuator target to hit the air bearing
stator.
A photograph of this setup is shown in Figure 10-5.
A bias current was
applied to the voice coil to preload the slide against the compression springs.
XM(s)II (s)
'
E
100
E
(D
5
E 10-5
10
102
103
102
10
0
(D
~j-200
o -400
D- -600-
-800
101
3
frequency [Hz]
Figure 10-4: Measured frequency response from voice coil current to air bearing
position.
We designed a lead-lag controller for the air bearing and targeted a crossover
frequency
f,
of 150 Hz. The lag compensator is a PI controller. We placed the PI
zero at a factor of eight below
fe,
or f
=
18.75 Hz. We designed for a phase margin
contributes
OM = 30'. The plant has a phase of -190.7' at 150 Hz. The PI controller
another 7.10 of phase lag at 150 Hz. Therefore, the lead controller must add 47.8'
369
Figure 10-5: Air bearing setup with preload springs.
of phase to achieve the desired phase margin. If we place the maximum phase of the
lead controller at the crossover frequency, we require a separation factor a between
the lead compensator zero and lead compensator pole of a = 6.72. This results in a
lead zero at
f
=
58.7 Hz and a lead pole at
f
= 389 Hz.
The gain of the plant at 150 Hz is 0.00868 mm/A. The gain of the lead com2.59 and the gain of the PI compensator at 150 Hz is
pensator at 150 Hz is \a
1.0078. Therefore, to achieve unity gain at a crossover frequency of 150 Hz, we select
K = 1/(0.00868 - 2.59 .1.0078) = 44.8 A/mm. The position feedback controller is
C. (s)
=
.K,
I a2
(8.488 x 10 3 s + 1
8.488 x 10-3s
2.752 x 10-3 s
4.092 x 10-4s + 1
The predicted controller and measured loop transmission frequency responses are
shown in Figure 10-6.
370
Position controller
-P
C
LT
100
E
10-5
101
103
102
200
-200
-
0
-
as -400
-C
-800
101
.
-
-600
103
102
frequency [Hz]
Figure 10-6: Air bearing frequency responses: P - plant (measured); C - controller
(predicted); LT - loop transmission (measured).
10.2
Actuator Calibration
We calibrated the reluctance actuator by mapping the current-gap-flux relationship
and force-gap-flux relationship. The position controller maintained a constant operating air gap during these calibrations. In the initial calibration routine, we drove
the reluctance actuator into positive and negative saturation at several different operating gaps with a 1 Hz voltage sine wave. We recorded the following measurements:
reluctance actuator current, sense coil voltage, load cell signal, air bearing position,
and voice coil current.
We recorded five to ten cycles of data. In post-processing,
we took the cyclic average of each measurement in an effort to reduce noise. We
integrated the sense coil voltage in post-processing to provide a flux estimate.
In
this initial calibration, we used only the position feedback controller to maintain the
constant operating gap. The block diagram for this setup is shown in Figure 10-7.
The variables highlighted in red are the recorded measurements. The variables I, and
v, refer to the reluctance actuator current and sense coil voltage, respectively.
371
R
v
_
C,
C
Position
control lert
tac
Actuator
Iv
vo~
eluctance
Fr
F
B
g-Enoe
Voice coil current
controller
tor.o
Figure 10-7: Controller configuration used for initial calibration of reluctance actuator.
Because the position feedback controller has finite bandwidth, there is some deviation from the nominal air gap during the calibration tests. Figure 10-8 shows the
cyclic air gap position during a calibration test at a nominal air gap of 500 Pm. The
reluctance actuator current during this test reached a peak value of 1.9 A and the
peak force as measured by the load cells was 150 N. The air gap is shown to have a
peak fluctuation of about
6 pm about the nominal. The peak gap fluctuation per
peak force is therefore 0.04 pm/N.
0.506
0.5040.502-
0.5
0.4980.4960.4940.492
0
0.2
0.4
0.6
0.8
1
time [s]
Figure 10-8: Air gap fluctuation during calibration at go
=
500 Jim.
After this first set of calibrations, we implemented two 2-D lookup tables (LUT).
We used the current-gap-flux relationship to generate a 2-D lookup table that estimated the reluctance actuator flux based on the actuator current and air gap. We
372
then designed a simple flux feedback controller using this estimated flux as the feedback variable. This flux estimate does not incorporate hysteresis. We used a second
2-D LUT to estimate the reluctance actuator force from the desired flux and air gap.
We used this predicted force to command a feedforward force to the voice coil actuator to improve the cancellation of the reluctance actuator force. This helped to
reduce the position deviation during the second set of calibrations and allowed us to
generate a more accurate reluctance actuator mapping. For this second set of calibrations, instead of commanding a sinusoidal voltage, we commanded a sinusoidal flux
density. The block diagram for this setup is shown in Figure 10-9. After generating
new calibration curves using this configuration, we could replace the B(I, g) LUT
and the F(Bd, g) LUT with more accurate mappings and run the calibration routine
again in an iterative process.
Bd
IReuctance
CB
Actuator
I'
B(I,,
,
Flux feedback
Ftux IU'
contoller,
2D LUT
controller i
I
controller
Current
feedback loop-
Position
feedback loop
Figure 10-9: Controller configuration used for calibration of reluctance actuator with
voice coil feedforward added.
Figure 10-10 shows the cyclic air gap position during a calibration test at a nominal air gap of 500im
with feedforward force control implemented on the voice coil.
The actuator current during this test reached a peak value of 2.9 A and the peak
373
measured force was 315 N. We were able to reach these higher force levels because
of the improved position control from the feedforward force. The air gap has a peak
1.6 pm about the nominal. The peak gap fluctuation per peak
fluctuation of about
force is therefore 0.005 pm/N, an eight-fold improvement over the calibration done
without feedforward.
0.502
-
0.501
,
0.5-
0.499-
0.498
0
0.2
0.4
0.6
time [s]
0.8
1
Figure 10-10: Air gap fluctuation during calibration with force feedforward at go
500 pm.
10.2.1
Flux-Current-Gap Calibration
Figure 10-11 shows the reluctance actuator B-I curves recorded at different operating
gaps using the calibration configuration of Figure 10-10. The operating gaps ranged
from 300 pm to 700 pm in increments of 50 pm. The 2-D LUT map generated from
these curves is shown in Figure 10-12. For current values beyond those measured in
the calibration routine, we extrapolated linearly to estimate the corresponding flux
density. At small air gaps, we could not drive the actuator very far into saturation
because the voice coil and amplifier could not sufficiently counteract the reluctance
actuator force.
For the LUT at these air gaps, we extrapolated linearly from the
measured current-flux points to generate missing table entries.
This is why some
of the flux density values at small gaps reach values greater than 2 T, which is not
physically possible for NiFe.
374
B v.
1.5
I
1-
0.5
----
S-
-350
-0.5-
-
-1
-
-1.5
-4
Ir
4
2
0
-2
300 Am
Am
400 Am
450 Am
500 Jim
550 pm
600 jm
650 pm
700 jm
[A]
Figure 10-11: B-I curves at different operating gaps for reluctance actuator calibration.
B(Ir ,g) LUT
,
4,
-
0,
-42
5
- 40.8
0.6
0
current [A]
-5
0.4
0.2
gap [mm]
Figure 10-12: B(I, g) lookup table.
Figure 10-13 shows the mutual inductance Lm of the reluctance actuator as a
function of the gap g. The experimental data points were obtained by computing
the slopes of the linear regions of the curves in Figure 10-11. A theoretical best-fit
curve proportional to 1/g is shown in red. This 1/g relationship is derived from (2.8).
Another fit is shown in blue, where we fit a function of the form
Lm3(g) =
(g + b)"
375
(10.3)
to the experimental data, where a, b, and y are the fitting parameters. We found that
the best fit for a function of this form to the experimental data was
1.22 x iO- 3
Lm (g) =
.2 x
n ,00377
(g - 0.0582)
(10.4)
where g is in units of mm.
3.5x1
-
0
3
experimental
free: 1/g"
theo: 1/g
2.5-
2-
1.5
0.3
0.4
0.5
0.6
average gap [mm]
0.7
Figure 10-13: Mutual inductance Lm of reluctance actuator as a function of gap g.
10.2.2
Force-Flux-Gap Calibration
Figure 10-14 shows the force-flux curves for the different operating gaps obtained from
the calibration routine. The force is measured by the load cells and the flux density
is computed in post-processing from the recorded sense coil voltage. As expected, the
relationship between the force and flux density is parabolic. The force decreases as
a function of gap. This is in contrast to the prediction from (2.13). The reason for
the difference is that (2.13) does not take into account fringing fields. As the air gap
changes, the fringing field patterns change. For example, as the air gap gets larger,
the fringing fields will take up a greater proportion of the total flux measured by the
sense coil. As a result, the force-to-measured-flux ratio will reduce.
The 2-D LUT generated from these curves is shown in Figure 10-15. The mapping
from flux density and gap to force is shown in the left plot. We also created the inverse
376
F v. B
40U
300-
20 0
300 pm
350 pm
0
400 pm
450 pm
500 pm
1C 0
550 pm
600 pm
650 pm
700pm
-1
0
-1.5
1
0.5
-0.5
0
flux density [T]
1.5
Figure 10-14: F-B curves at different operating gaps for reluctance actuator calibration.
2-D LUT from these curves in order to generate a desired flux density from a desired
This inverse LUT is represented as the B(Fd, g) block in Figure 10-1.
force.
The
inverse map is shown in the right plot of Figure 10-15.
B(F,g) LUT
F(B,g) LUT
1.5,
600,
500,
F400
0.5,
- 300,
200,
01
-0.5;
400
10u
0.7
1
-1
flux density [T
300
0.6
0
-1
-2
0.4
0.3
201
200
0.5
force [N]
gap [mm]
0.5
0
0
gap [mm]
Figure 10-15: LEFT: F(B, g) lookup table; RIGHT: Inverse B(F, g) lookup table.
377
The force can be expressed as a function of the flux density in the form of
F = k(g)B 2
,
(10.5)
where k(g) is called the k-factor and is a function of g. The k-factor thus captures the
gap dependency that is absent from (2.13). In Figure 10-16, we plot the k-factor as a
function of gap as determined from the calibration data. From (2.13), the predicted
k-factor (gap-independent) is k = A/(2po) = 289 N/T 2 . The measured k is lower
than the predicted value because of the effect of fringing fields. If we fit a function to
the measured k values, we find that
-
179
(10.6)
k(g) = (g
_ N/T02,
(g + 0.32)0.48
where g is in units of mm. We find that the gap dependency of the k-factor follows an
approximately inverse square-root law. We use this function to extend the F(B, g)
LUT and B(F, g) LUT shown in Figure 10-15 beyond the range of the data we
collected.
k-factor
260
o
experimental
fit
240
K 220
200
180
160'
0. 2
0.3
0.4
0.5
0.6
average gap [mm]
0.7
0.8
Figure 10-16: Reluctance actuator k-factor.
378
10.2.3
Voice Coil Calibration
We calibrated the voice coil current against the load cell measurements in order to
generate the voice coil feedforward current.
The voice coil current I,, is plotted
against the reluctance actuator load cell force measurement Fr in Figure 10-17. The
data is taken from the calibration test done at go = 500 pm. A linear fit reveals the
slope to be 0.0329 A/N. This is the value used for the k-j block in Figure 10-1 and
Figure 10-9. The motor constant Kmvc is the inverse of this and is estimated to be
30.4 N/A. The specification sheet lists the motor constant as 35.14
10% N/A [10].
Our estimated motor constant is slightly below the minimum.
There is also a small offset of -30 mA in the voice coil current at zero reluctance
actuator force. It is believed that there is a small attractive force between the voice
coil magnet assembly and a long steel bolt used to attach the coil assembly to the
mounting bracket. Further evidence for this hypothesis is that with power to the air
bearing setup shut off, the air bearing slide moved toward the voice coil mounting
bracket. The position controller compensates for this attractive force with a DC offset
force. This accounts for the voice coil current offset. This offset was added to the
feedforward control.
10.3
SHM Observer Implementation
With the actuator feedforward calibrations completed, we then implemented the SHM
observer described in Section 6.6.
The observer structure is shown again in block
diagram form in Figure 10-18. In the block labeled 'cg 2 ', we have replaced 2/po with
the constant c. This signifies that we can fit the constant c to experimental data
to generate a better fit. We first fit the SHM to experimental data recorded at a
single air gap. We then verified the SHM in real-time under several different testing
conditions. We included the feedforward path in Figure 10-18 for better performance.
379
Ivc v. Fr
12
101F
8
6[
0i
0
.5 4
0
2
0
-
-2
100
0
measu red data
linear f it
400
300
200
load cell force [N]
Figure 10-17: Plot of voice coil current versus reluctance actuator load cell measurements used to estimate Kmvc.
Feedforward Path
/ACLUator
Mvodel
'Fe
NI
+
+
C+
N!
2+
2
B
Controller
B(H2cg
Sheared
Hysteresis Model
-- - - g
- -g9
Figure 10-18: Observer block diagram with SHM.
10.3.1
Calibration
We calibrated the SHM at a nominal operating gap of 530 jim against the sense
coil measurement.
The reluctance actuator was driven into positive and negative
380
saturation with a 1 Hz voltage sine wave. The position feedback controller with voice
coil actuator maintained the constant operating gap. Feedforward control with the
voice coil actuator was included to reduce deviations from the nominal operating gap.
The reluctance actuator current and sense coil measurements were recorded. In
postprocessing, cyclic averages over five cycles of the current and sense coil data were
computed, and then based on (2.35) the averaged sense coil data was integrated and
scaled by 1/(NAp) to give a cyclic average of the flux density B. This data is shown
in Figure 10-19. The Preisach model was fitted to the resulting B-I major loop data
using the Hui method described in Section 3.3.2.
B v. I
1.5
1-
E
0
-0.5
[
-1._4
-3
-2
0
-1
1
2
3
4
current [A]
Figure 10-19: B-I data at 530 pm gap used to calibrate SHM.
The constant c in Figure 10-18 can be determined by inspecting the linear portion
of the B-I curve. Since c replaces 2/po in the block diagram of Figure 10-18, we can
substitute c for 2/po in (2.8) and write
B= NI
cg
(10.7)
NI
Bg
(10.8)
Solving for c, we write that
C-
If we express B as B
=
mI, where m is the slope of the B-I curve when g = go, we
381
can write
C= N
(10.9)
mgo
The slope m of the linear portion of the B-I curve was found to be 0.427 T/A. This
results in c
=
10.3.2
Observer Controller Design
1237
A . The parameter go for the SHM is set to go = 0.530 mm.
T-mm~
We selected an integrator as our observer controller. The controller gain K was set
such that the observer crossover frequency
f, would
be 2 kHz at the nominal gap.
From (6.34), we set
Ko-
f
- 1.26 x 105 m-Is- 1,
(10.10)
lFe
and our observer controller is
Co(s) = 1.26 Sx 0
10.3.3
(10.11)
Experimental Verification
To verify the SHM, we implemented the SHM observer in the dSPACE RTC. We used
an embedded Matlab [60] code to implement the Preisach model.
For testing, we used the position feedback controller to keep the air gap at the
desired gap, whether a constant gap or an applied gap disturbance. The reluctance
actuator was then driven with a voltage profile, and the SHM observer estimated the
flux density output in real-time from the current and encoder measurements.
We
integrated the sense coil measurement in post-processing to give us a flux density
estimate from the sense coil and then compared this result to the real-time estimate
from the SHM.
Major Loop at Nominal Gap
Figure 10-20 shows the B-I major loop at nominal gap of the observer real-time
flux estimate compared to the integrated sense coil measurement.
382
The sense coil
measurement is shown in blue and the observer estimate is shown in red. The fit
between the two is excellent and shows good agreement both near the origin and near
saturation.
B v. I
1
1
2.8
2.9
1.3
sense coil integrated
hysteresis estimate
-
1.51111
-
-
0.5
2.7
3
3.1
3.2
3.3
-
2.6
0.05
-
-0.5
-
0-
1
-1.5
-4
-2
0
I[A]
-0.05
-0.5
4
2
-
-1
0
I [A]
0.5
Figure 10-20: LEFT: major loop of SHM observer real-time flux estimate compared
to sense coil measurement at nominal gap of 530 pm; TOP RIGHT: zoomed in at
saturation; BOTTOM RIGHT: zoomed in at origin.
Minor Loop at Nominal Gap
Figure 10-21 shows the observer real-time flux estimate compared to the integrated
sense coil measurement for a B-I minor loop centered about the origin at the nominal
gap. The fit between the two is reasonable but is not as good as it was for the major
loop. This is because in identifying the hysteresis model, only the major loop data
was used. An implementation of the Preisach model that takes into account minor
loop data during calibration could possibly provide a more accurate fit. Nevertheless,
the hysteresis model still performs better than a single-valued nonlinear estimate.
The single-valued flux density estimate b is defined as
f
=
2
B[ B(I) +Bd(I)] ,
(10.12)
where Ba is the flux density of the ascending branch of the major loop, and Bd is the
flux density of the descending branch of the major loop. The RMS error between the
383
observer estimate and the sense coil measurement is 2.2 mT, compared to 12.6 mT for
the single-valued estimate.
B v. I
-
0.5
0.45
-
--
0.4-
sense coil integrated
-hysteresis estimate
0.3-
0.4
0.2
0.35
0.1
0 .7
0
0.8
0.9
1
1. 1
0
0.1
0.2
-
7
0.05
-0.1
-
-0.2
0
-0.3-
[
-
-0.4
-1.5
-1
-0.5
0
I[A]
0.5
1
1
-A- nr
'
'
-
-0.5
1.5
-0.2
-0.1
I
[A]
Figure 10-21: Left: minor loop of SHM observer real-time flux estimate compared to
sense coil measurement at nominal gap of 530 pm; Top right: zoomed in at maximum
flux density; Bottom right: zoomed in at origin.
Gap Offset from Nominal Gap
We next changed the air gap from 530 pm to 580 pm. Figure 10-22 shows the resulting
B-I major loop of the observer real-time flux estimate compared to the integrated
sense coil measurement.
The observer flux estimate shows good agreement with
the sense coil measurement.
There is some discrepancy at saturation.
The RMS
error between the observer estimate and the sense coil measurement is 5.8 mT, while
the RMS error between a single-valued estimate and the sense coil measurement is
18.6 mT.
One of the potential error sources is that our lumped parameter model does not
take into account fringing flux, which will change with air gap.
The fit could be
improved by calibrating the actuator at multiple gaps, determining the relationship
between B and g, and then changing the gap dependency in (6.30) accordingly. We
can generalize the SHM to other gap dependencies. Suppose that we express (4.1) as
lFeH + f(g)B
384
=
NI,
(10.13)
B v.
1
1.5 1
I
1.3-
sense coil Integrated
estimate
-hysteresis
L
-
1
1.251.2
1.15
0.5 F
2 .8
2.9
3
3.2
3.1
3.3
3.4
3.5
0
0.05
-
-0.5
-
0
-1
-1.5-4
-2
-0.05_0.5
4
2
0
1 [A]
0
1 [A]
0.5
Figure 10-22: LEFT: SHM observer real-time flux estimate compared to sense coil
measurement at gap of 580 lim; TOP RIGHT: zoomed in at saturation; BOTTOM
RIGHT: zoomed in at origin.
where f(g) is a function describing the gap dependency. We can follow a similar procedure to the change-of-variables procedure used in deriving the SHM in Section 6.6.
However, we now define the variables
f2
=
f2
and H2 as
f(go),
(10.14)
H + f(go)B
(10.15)
f(g) -
and
H2
lFe
We can then express (10.13) in terms of f2 and H2 as
lFe (H
2
-
lFeH2
+ (f2 + f(go)) B
= NI,
- f(go)B + f 2B + f(go)B
= NI,
f( 9 0 )B
IFeH2 + f 2 B =
NI.
(10.16)
This is the same result that we obtained before in deriving (6.30), except that now we
385
use f2= f(g) - f(go) instead of g 2 = g - go. The B(H2 ) hysteresis model is calibrated
the same way as was done before: we measure the B-I relationship at the nominal
gap where f2= 0. For f(g), we implemented the relationship given in (10.4) in our
SHM observer in an attempt more accurately to model the gap dependency of the
reluctance actuator. However, while simulations indicated that the we were able to
implement this relationship successfully, the dSPACE RTC would crash shortly after
compiling. Time constraints precluded us from investigating this further.
Another possible error source is that the actuator stator heats up as current is
driven through it. This causes the stator to expand, resulting in the air gap becoming
smaller. This air gap change is invisible to the encoder, so the observer model will
not accurately account for it. We discovered this by clamping the air bearing slide to
the reluctance actuator stator with a plastic shim between the stator and target. We
then drove the actuator with a 1 Hz voltage sine wave. The actuator current reached
an amplitude of 3.2 A, so the RMS power dissipated is 1/2 x (3.2 A) 2 x 1.5Q = 7.7W
The encoder measurement during this test is shown in
for a 1.5 Q coil resistance.
Figure 10-23. We can see that the encoder measurement slowly increases over time.
Over ten seconds, the average position changed by ~100 nm.
0.5198
0.5196
0.5194
0,5192
0
0.519
0.5188
0.5186
0.5184
0
1
2
3
4
5
6
7
8
9
10
time [sec]
Figure 10-23: Encoder measurement showing reluctance actuator stator expansion
due to actuator heating.
We ran a longer-term test where we drove the reluctance actuator with a con-
386
shim
stant DC voltage with the stator clamped to the air bearing slide with a plastic
meabetween the target and stator. We measured the resulting current and encoder
surement. The current gradually decreases as the coil resistance increases owing to
the coil heating. The current and encoder measurements are shown in Figure 10-24.
The current measurement shows a sudden decrease around 1700 seconds. It is unclear
what the cause of this is, although it is thought to be a problem with the measurement
as there seems to be no effect on the position measurement at this time. The current
measurement also becomes much noisier after this point. The encoder measurement
shows an increase of over 20 im during the test, demonstrating that actuator heating
can have a significant effect on the air gap. The nominal power dissipation during
2
this test was about (3.2 A) x 1.5 Q = 15.4 W. In the real application, the actuator
will be water-cooled, so expansion from heating will be much less of a problem.
Encoder measurement
Current
0.53
4
~0.525-
3
3
-
S0.52
2
cL
0.505-
00
0.515
1000
2000
3000
0
4000
time [sec]
1000
2000
3000
4000
time [sec]
Figure 10-24: Reluctance actuator driven with constant voltage and air bearing slide
clamped to stator. LEFT: reluctance actuator current measurement. RIGHT: encoder measurement.
Sinusoidal Gap Disturbance
We used the voice coil to apply a 4 Hz gap disturbance with 11 pm amplitude about
the nominal operating gap of 530 pm.
1 Hz voltage sine wave.
The reluctance actuator was driven with a
Figure 10-25 shows the resulting B-I major loop of the
observer real-time flux estimate compared to the integrated sense coil measurement.
The observer estimate shows excellent tracking with the sense coil measurement in
the presence of the sinusoidal gap disturbance.
387
B v. I
1.5
-
I
sense coil integrated
hysteresis estimate
1
0
0.51
-0.05
-0. 5
0.5
-1.1
-0.51
-1.2
-1
-1.3
1.
5'
-4
-2
0
-3.3
4
2
I[A]
-3.2
-3.1
-3
-2.9
-2.8
-2.7
I [A]
Figure 10-25: LEFT: SHM observer real-time flux estimate compared to sense coil
measurement with sinusoidal gap disturbance about nominal gap of 530 1m; TOP
RIGHT: zoomed in at origin; BOTTOM RIGHT: zoomed in at saturation.
Scanning Gap Disturbance
We applied a gap disturbance to the air bearing slide that was similar to a typical gap
disturbance on a real lithography stage during scanning. Figure 10-26 shows the gap
disturbance that was applied to the air bearing stage. The peak-to-peak amplitude
is about 35 Jxm.
0.5 5
0.5
E
-.-
0.5 4
3
- -.
-
--
-
--
0.5
_
L _ _ I_ _ _
0.51
_
0
__I
_
0.2
__I
I
0.4
0.6
0.8
1
time [sec]
Figure 10-26: Gap disturbance applied to air bearing stage.
Figure 10-27 shows the B-I loop of the observer real-time flux estimate compared
388
to the integrated sense coil measurement in the presence of this gap disturbance. The
reluctance actuator was driven with a 1 Hz voltage sine wave. The observer estimate
shows good agreement with the sense coil measurement in the face of the dynamic
gap disturbance.
B V.
-
I
.
- 6d
1.3
sense co nerae
I- -hysteresis
estimate
I
0.5
1.2
F
2.6
2.7
2.8
2.9
3
3.1
3.2
0.05
-0.5 F
E
0-.
-1
54
-1 .-
-2
0
I [A]
2
-0.051
-0.5
4
0
I [A]
0.5
Figure 10-27: LEFT: SHM observer real-time flux estimate compared to sense coil
measurement with scanning gap disturbance about nominal gap of 530 pm; TOP
RIGHT: zoomed in at saturation; BOTTOM RIGHT: zoomed in at origin.
10.4
Hybrid Flux Estimator
For accurate high-bandwidth flux feedback control, we need a way to estimate the
flux for frequencies from DC to several kHz or beyond. The flux estimation needed
for the flux controller is represented by the 'Flux Estimation' block in Figure 10-1.
For this flux estimation, we combined the sense coil measurement and the SHM approximation based on the current and gap measurements: the sense coil provides an
accurate feedback signal at higher frequencies, while the current and gap measurements provide an accurate feedback signal at lower frequencies including true DC.
Some preliminary results with this scheme using a lookup table instead of a hysteresis model can be found in [5, 4]. Figure 10-28 shows a complementary filter structure
used for combining the two signals.
The sense coil measurement is passed through a second-order high-pass filter
389
Hysteresis
model
Low-pass filter
Sense
Resistor
>
.
Gap
LF
High-pass filter
Sense
HF
Coil
Figure 10-28: Complementary filter structure for estimating actuator flux.
(HPF) and then integrated, resulting in the signal
AHF.
The filter is second-order
rather than first-order so that any offset in the flux-rate measurement is not transferred to the high-frequency flux estimate after integration. To understand this, we
write the transfer function from the sense coil measurement v. to high-frequency flux
estimate AHF using a first-order high-pass filter as
AHF(s)
Vs(s)
=
HPF(s) --
s
S
i
S + WF S
AHF(S)
V(s)
8s
1
=
S + WF' ,+W
(10.17)
where WF is the cutoff frequency of the filter. The transfer function has a DC gain
of 1/WF, so any voltage offset from the sense coil buffer will result in a DC error in
AHF.
If instead we use a second-order filter, we write
AHF(s)
V8(s)
s2
(s
+wF)2S
AHF(S)
S
Vs(s)
(s + WF)2
390
The resulting transfer function now has a DC gain of zero, so any voltage offset at
the input will not propagate to the output. For our complementary filter pair, we set
WF
to 6.28 rad/s, which corresponds to a cutoff frequency of 1 Hz.
The current sense resistor and encoder measurements are passed into a hysteresis
model that estimates the flux as its output. In our setup, the SHM observer described
in the previous section is used as the hysteresis model. The output of the hysteresis
model is passed through a complementary low-pass filter (LPF). The low-pass filter
is defined as
LPF = 1 - HPF,
(10.19)
so that the resulting complementary filter pair sums to one. In our case, using (10.18)
for the high-pass filter, we can write the low-pass filter transfer function as
=1-
,
LPF(s)
+
(s
WF)2
(s+WF
2
2
(S -+WF)
s2+2
'
(S-+WF)
_2 _
2
+WF _ S2
2
(S+WF)
LPF(s)
=
2WpS + wF
(s -- WE) 2
We see then that the low-pass filter has a zero at
signal
ALF
is summed with
AHF
(10.20)
.
WF/2.
The resulting low-pass
to provide a flux estimate for feedback control that is
accurate at both low and high frequencies. With this scheme, we take advantage of the
benefits of a sense coil over other types of flux sensors, such as a Hall sensor, while the
hysteresis model compensates for the sense coil's poor performance at low frequencies.
Because the sense coil provides an accurate measurement at high frequencies, the
hysteresis model only needs to be accurate at low frequencies.
flexibility.
This gives us more
For example, we can separate the hysteresis model computation from
the sense coil measurement: the computation of the hysteresis model could be done
391
with a real-time computer at a relatively low sample rate, but then we could use a
high-bandwidth analog controller or FPGA to run the flux control loop.
10.5
Flux Control Loop Design
Using the calibration results and the hybrid flux estimator, we designed the flux
controller represented by the dotted red box in Figure 10-1. A block diagram of the
flux controller is shown in Figure 10-29.
Plant
P1 Controller
Bd
g
S
B
B
sBB
VEMF
........................................
_
Flux Estimator
g
Figure 10-29: Block diagram of flux control loop.
The plant is described by (4.52), repeated here as
Vs= IR+NAP
dt
.
(10.21)
If we linearize the plant about an operating gap go as was done in (4.54), we can
write the transfer function from Vs to B as
B(s)
= poN
Vs(s)
2go
1/R
s +1
_
1
L
NAP Ls + R'
where L is the actuator inductance. At low frequencies, B(s)/Vs(s) ~ L/(NApR).
Because L depends on g, the DC gain of the transfer function will vary with nominal
392
gap. At high frequencies, B(s)/Vs(s) ~ 1/(NAps), and there is no dependence on
gap.
If we used a high-bandwidth current control inner loop instead of driving the
reluctance actuator with a voltage, then our plant could be approximated from (2.8)
as
B = ONI
(10.23)
2g
In this case our plant will depend on the operating gap at both low frequencies and
high frequencies.
Therefore, one benefit of driving the reluctance actuator with a
voltage instead of a current is that if the flux loop crossover frequency is set much
higher than w
=
R/L, the crossover frequency and phase margin will remain constant
irrespective of gap, thereby ensuring stability over a wide range of gaps. This permits
the reluctance actuator to be used for stable position control. Experiments demonstrating position control with the flux-controlled reluctance actuator are documented
in [4].
Figure 10-30 shows the measured frequency response of the reluctance actuator
from input amplifier voltage to output flux linkage as measured by the sense coil.
The frequency response therefore includes any amplifier dynamics. We also had a 10
kHz low-pass filter on the sense coil differential amplifier channel, so the effect of this
filter is also present in the frequency response. The air gap was set to 500 1m. The
frequency response was acquired with the dSPACE DAQ, where we set the sampling
frequency to 40 kHz. The variable A is obtained in post-processing by multiplying the
magnitude of the measured sense coil variable dA/dt by 1/w, where w is the frequency
of the signal, and subtracting 90* from the phase of dA/dt.
We designed our controller based on the actuator frequency response. To express
the actuator plant in terms of B rather than A, we divided the frequency response
magnitude by NAp, where N, is the number of sense coil turns. We initially targeted
a crossover frequency
f,
of 3.65 kHz. At this frequency, the plant phase is -143'
the gain (in terms of B rather than A) is 0.001325 T/V.
393
and
To achieve this crossover
6 (s)/ V(S)
C'j
E
10~4-
.a10-6
E
10
102
103
104
10
102
103
10
) -100
C,)
-
-200 -
4
frequency [Hz]
Figure 10-30: Measured frequency response for reluctance actuator.
frequency with sufficient phase margin, we use a PI controller of the form
CB(s) = KB
TBS -I
1
(10.24)
-
TBS
We set the frequency of the zero to a decade below the intended crossover frequency, so
that the zero frequency
f,
= 365 Hz. From this, we set TB = 1/ (27fz)
=
4.356 x 10-4 S.
The gain of the PI compensator at 3.65 kHz is 1.005. To achieve a unity gain at 3.65
kHz, we set KB = 1/(0.001325 - 1.005) = 750 V/T.
After implementing this controller and inspecting the experimental loop transmission, we modified our controller to be slightly more aggressive and targeted a crossover
frequency of 4 Khz. We increased our controller gain to KB = 834 V/T. Our final
controller is
.
10-S
4.356 x i04s +1 1
.
CB(s) = 834 4.356 x 10--4
4.356
(10.25)
Figure 10-31 shows the measured loop transmission frequency response with this
controller.
The measured plant frequency response B(s)/V(s) and the predicted
394
controller frequency response are also shown.
The small 'blips' on the measured
loop transmission are the result of the controller signal saturating.
The input was
then reduced at these frequencies and the frequency response was retaken from that
The frequency responses with different input amplitudes were then
point onward.
concatenated together to cover the entire frequency range.
The loop transmission
shows a crossover frequency of 4 kHz with a phase margin of 30'.
Reluctance actuator flux controller
105
ca
777
10 0
1IT
105 ~1
10
102
103
1C
10 3
1
0
-50
:0 -100
-7,7
(D
0)c -150
-200
-250
Uc
1
102
frequency [Hz]
-
Figure 10-31: Reluctance actuator frequency responses: P - plant (measured); C
controller (predicted); LT - loop transmission (measured)
10.6
Force Control with Flux Feedback
We tested the reluctance actuator force accuracy with the flux feedback loop in place.
We measured the force frequency response of the actuator and then measured the stiffness frequency response. We also measured the force accuracy of the actuator when
trying to follow a desired force pulse profile. The block diagram for the reluctance
actuator force controller is shown in Figure 10-32. A desired force Fd is sent to the
inverse 2-D LUT described in Section 10.2.2 to generate a desired flux density Bd.
395
The flux control loop drives the reluctance actuator to generate a flux density B that
tracks Bd. The resulting actuator force is measured by the load cells. The desired
force is also fedforward to the voice coil actuator as shown in Figure 10-1.
(g
Flux feedback
loop
Lookup Table
Force Measured
- - - - - - - - - - - - - - - - - - -- - with Load cells
FaB+Reluctance
B(F1,)
BCB
Actuqator
B
FBg
F
-
Desired
force
IN cont-rore-rI
Flux
Estimation
g
g
Figure 10-32: Block diagram for reluctance actuator force control.
10.6.1
Force Frequency Response
We measured the frequency response from Fd to F while the air gap was maintained at
530 pm with the position controller. We measured frequency responses for bias forces
FO of 5 N, 25 N, and 50 N. Figure 10-33 shows the measured frequency responses of
F(s)/F(s). We were unable to measure frequency responses at higher bias forces
near saturation because the voice coil actuator will overheat at the high continuous
currents required to offset the reluctance actuator bias force. The amplifier is also
limited in the amount of continuous power it can deliver to the voice coil.
The measured force tracks the desired force up to about 100 Hz.
Beyond this
frequency, the performance deteriorates and we see various resonances and antiresonances.
These are likely the result of structural resonances affecting the load
cell measurement. Moreover, the air gap is not perfectly constant because of the finite bandwidth of the position controller. The position controller has a loop crossover
of 150 Hz, so beyond this frequency, the imperfect gap disturbance rejection of the
force controller will also affect our accuracy. Acceleration of the optical table will also
result in errors in the load cell measurement. These errors become more pronounced
at higher frequencies, owing to the w2 dependency of acceleration.
396
F(s)IFd(s)
10
100
0
E 10-
10
t) _2 0)
0 --0-200
O -400 -
a.
-600 -0
-
102
-
103
104
-
200
10
F
=5-
____F0 =5N
F =25N
F =50 N
-800
101
102
10
3
10
frequency [Hz]
Figure 10-33: Measured flux-controlled reluctance actuator frequency responses from
desired force to measured force.
Figure 10-34 shows the frequency responses for frequencies below 100 Hz.
through 20 Hz, the measured force magnitude remains within
2%
Up
of the desired
force magnitude. Part of the inaccuracy may be the result of the limited load cell
resolution. This resolution can be changed on the charge amplifier. However, when
we used a higher resolution setting, the amplifier would quickly overload before much
data could be captured. The spikes that are circled in the magnitude plot are artifacts:
these occurred when the charge amplifier overloaded and had to be reset. The phase
shows a gradual increase with frequency. Likewise the magnitude begins to increase
after about 20 Hz. This is thought to be due to the loop-widening phenomenon present
in the sense coil measurements, resulting in the phase of the measured force leading
the phase of the measured flux.
397
F(s)/Fd(s)
1.05
C
1
E
0.95
3 -o,2
CL
-F
10
10
101
102
FO=5 N
F =25 N
= 50 N
1
0
-1
frequency [Hz]
Figure 10-34: Measured flux-controlled reluctance actuator frequency responses from
desired force to measured force for frequencies below 100 Hz.
10.6.2
Reluctance Actuator Stiffness
The reluctance actuator stiffness was measured by using the flux controller to maintain a constant force, applying a gap disturbance using the voice coil actuator, and
measuring the resulting force disturbance.
We first present stiffness results of the
flux-controlled actuator with the SHM observer incorporated into the hybrid flux
estimator. We then present the results of a previous set of experiments done with
a flux-controlled actuator but using the Chua model instead of the SHM. We then
present a theoretical analysis of the effect that imperfect feedforward and sampling
delay have on stiffness.
Stiffness Results with SHM
Figure 10-35 shows the measured frequency responses of the reluctance actuator stiffness for bias force levels of 5 N and 35 N. The stiffness for both bias levels is shown to
be well below the maximum allowable stiffness, demonstrating that the flux controller
398
is effective in rejecting gap disturbances. For example, the stiffness of the reluctance
actuator at 10 Hz is 0.003 N/pm. This results in a force error of less than 0.01% of
the full-scale force for a 10 1m air gap disturbance at this frequency. The maximum
allowable stiffness is based on the 99.9% feedforward force accuracy specification for
the reluctance actuators. For our prototype actuator, the maximum force is approximately 400 N.
The maximum force error permitted is therefore AF = 0.4 N. A
typical gap disturbance is on the order of Ag = 10 11m.
Therefore, the maximum
allowable stiffness kr,max is estimated to be
kr,max -
Ag
= 0.04 N/pm.
= 0
AF
10pJm
_0.4N
(10.26)
This value corresponds to the horizontal dotted green line in the figure. The majority
F(s)/g(s)
E 100 - Max allowable stiffness
- - ---- 00
- - ----10101
-
- -
1o
4
-
1000
103
102
10
=5 N
FO = 35 N
0~FO
0
-1000
c
-C
-2000
-3000
-4000
101
102
103
104
frequency [Hz
Figure 10-35: Measured frequency responses of flux-controlled reluctance actuator
stiffness.
of the frequency content of a typical gap disturbance during scanning is below 30
Hz. Gap disturbance data from ASML was analyzed in [62], and the gap disturbance
amplitude was found to fall off as approximately 1/w at higher frequencies. The max-
399
imum allowable stiffness requirement is therefore less stringent at higher frequencies,
and is shown in the figure as the green dotted line with positive slope. This line has
a slope of w.
For frequencies below 20 Hz, the measured stiffness decreases with frequency. This
may be because the signal-to-noise ratio from the load cell measurement is low at these
frequencies. For frequencies beyond 20 Hz, the stiffness increases with approximately
a slope of w. This matches the the theoretical prediction of (4.64) with an integrator
in the controller. At low frequencies, the stiffness also increased with bias force as
expected.
However, at higher frequencies, the bias force did not seem to have an
effect. This may be the result of the force feedforward, which if imperfect will also
introduce a stiffness. Using (4.64), we predict a stiffness of about 4 x 104 N/km
at 100 Hz for FO = 35 N. From the measured frequency response in Figure 10-35,
we find a measured stiffness of 0.0065 N/pm, a factor of 16 greater than predicted.
This supports the hypothesis that imperfect feedforward is dominating the actuator
stiffness.
Stiffness Results with Chua Model
In a previous set of experiments before we had developed the SHM, we measured
the stiffness of the flux-controlled actuator using the Chua model in the hybrid estimation scheme. The flux controller was nearly identical to the one that was described in Section 10.5, however, and we achieved a similar loop crossover frequency
of 4 kHz. Figure 10-36 shows the measured disturbance rejection frequency responses
(B(s)/g(s)) of the flux loop alone (without the force inversion) for different levels of
bias flux density BO on the reluctance actuator prototype. The nominal air gap was
500 prm. The BO levels of 0.09 T, 0.28 T, and 0.62 T correspond to bias force levels
of 1.5 N, 15 N, and 75 N, respectively. As predicted in (4.61), the magnitude of the
frequency response increases with BO.
Figure 10-37 shows the measured disturbance rejection frequency responses for
different nominal air gaps. The bias force was 1.5 N. As predicted by theory (4.61),
the frequency response does not depend on go.
400
B(s)/g(s)
as
1o3
102
101
1o
4
400
200
1
0B0 =O.09 T
-600..........I
1-200
O-400
-400
101
=0.28 T 1B
B =0.62 T
6
1
102
3
1=
frequency [Hz]
Figure 10-36: Measured frequency responses of flux-controlled reluctance actuator
gap disturbance rejection at different flux levels.
Figure 10-38 shows the measured frequency responses for the compensated actuator stiffness at two different nominal operating air gaps. The bias force is 35 N in
both cases. We can see that there is not much discrepancy between the two, illustrating that the nominal air gap does not have an effect on the stiffness. The break
in the plots at 100 Hz is due to recording the frequency responses in two steps: at
the lower frequencies, a higher gap displacement was used (25 Pm amplitude), while
at the higher frequencies, a lower gap displacement was used (5 pm amplitude). This
was done to avoid overdriving the voice coil actuator.
The stiffness is higher than
what was shown for the FO = 35 N plot in Figure 10-35. This indicates that the SHM
helps significantly in reducing the actuator stiffness.
Stiffness from Imperfect Feedforward
In this case of perfect inverse force-flux-gap modeling (B(F,g)) and no gap sampling
delay, the only stiffness in the compensated actuator results from the finite bandwidth
401
B(s)/g(s)
E
100~
10-5 --
10
12
10
102
10 3
10
102
10 3
10 4
200
0
-
-2D
i-200
o = 400
pro
go = 500 pm
0L -400 -=
600 pm
-001
10
frequency [Hz]
Figure 10-37: Measured frequency responses of flux-controlled reluctance actuator
gap disturbance rejection for different nominal air gaps.
of the flux control loop. In reality, the force-flux-gap relationship will not be modeled
perfectly. Since the compensated stiffness in Figure 10-35 is worse than we would have
predicted if the flux loop finite bandwidth were the only contribution to stiffness, we
expect that imperfect feedforward is dominating our stiffness. Using (10.5) to express
the force, we can write the change in force AF about an operating point as
AF
OFAB +
Ak(g) = 2koBoAB + Ak(g)B2,
Ok
OB
(10.27)
where Bo and go are the flux density and air gap at the operating point, respectively,
and where ko = k(go). With reference to the block diagram of Figure 4-16, we can
approximate AB as
AB ~ GdABd + GdAg,
(10.28)
+
where G,1 is the closed-loop transfer function of the flux loop, equal to C(s)P(s)/(1
C(s)P(s)), and Gd is the closed-loop disturbance rejection transfer function, equal to
402
F(s)/g(s)
105
E 1005
z10
~
10 _
104
102
103
104
102
103
104
0
)
500
ngo
-c
-1000
= 500 pm
g = 600 pm
-1500
101
frequency [Hz
Figure 10-38: Measured frequency responses of flux-controlled reluctance actuator
stiffness at different nominal air gaps.
D(s)/(1 + C(s)P(s)). We can related the desired force Fd to B as
(10.29)
Fd = k(g)B ,
where k(g) is the feedforward approximation of k(g). Solving this for Bd, we write
that
.30)
B=
(10
k(g)
Taking partial derivatives again, we can approximate ABd as
ABd
2'B
&ik
Ak(g) =
11fd
k
2k
0
- Ak(g),
k0
(10.31)
where ko = k(go). We have ignored any contribution from AF since we are concerned
with the response in AF to a gap disturbance rather than a change in the desired
403
force. We substitute (10.31) and (10.28) into (10.27). We then write
AF
2kOBO [GeIABd+Gdg]
1
+ Ak(g)BO,
)+Ak(g)B.
FA
(10.32)
If we divide both sides by Ag and take the limit as Ag approaches zero, we can derive
an expression for the compensated stiffness kc = OF/ag. This yields
krcF
krc = 2koBo
1Gik
We can note from (10.5) that BO =
Fd =
+ G
a+ -- B .
(10.33)
/FO/ko, where F is the force at the operating
FO, we can simplify (10.33) to
krc
krc
kkc=
F0
11
-Ge k0
2 k0
-2ko
k-
F0
F0 O
-N
+ 2koGd
k 0Og
8kF
k- 0 &
ag +k2or kF
ko V/koko
F0
&k F0
+
k0
Og ko'
-
0 &ok
g+
ko ag,
,
point. If
18
(10.34)
This is a general expression for the compensated stiffness that includes both the effects
of imperfect feedforward and finite flux loop bandwidth. If ko = ko, we can further
simplify this expression to
krc =
N + 2Gd
F0 (Ok
Of
ko (
ag)
g
/koF0 .
(10.35)
Thus, if the feedforward is perfect and Ok/Og = Ok/Og, the compensated stiffness
reduces to 2Gd\/kOF. This result is consistent with the compensated stiffness derived
in Chapter 4. We can use (10.35) to investigate the effect that different k-factors and
different types of mismatches between k and
the stiffness.
404
k (e.g., offset error, gain error) have on
Effect of Sampling Rate on Stiffness
Suppose that our force feedforward is perfect, that is, k(g) = k(g). The fact that the
gap measurement will be delayed by one time step will still introduce a stiffness into
the system. Assuming that we have a perfect flux feedback loop (B = Bd), we can
write F as
F = k(g)B 2 = k(g)Bd =
k(g - Ag)
F,
(10.36)
where Ag is the difference between the actual gap and the measured gap as a result
of the sampling delay. The force error AF = F -
AF
k(g)
k(g - Ag)
=F k)
Fd
is
Fd- Fd,
Ag)~
- [k~g)(10.37)
- k(g
d
k(g - Ag)_
1.7
Expanding k(g - Ag) into a Taylor series, we can write
~~ F.
AF
d
k(g) - k(g) + lAg
k(g) )gg
2Ag
~Fg
AF
~
Fdk(g)
- 'Ag
d
4g
k(g) 1 -
.(10.38)
ILkAg'
Since the force-flux relationship dependency on gap is a second-order effect, then
for small gap disturbances, we can say that 1/k(g)(ak/&g)Ag << 1, and we can
405
further approximate (10.38) as
AF
Fd okA
1 -Ok
k(g)ag
k(g)og g
k Ag +
I1
k(g)
g59
k
k(g)g
)
'
Ag
(+Ag
Fd kAg.
k(g) Og
(10.39)
In the last line, we have ignored the second-order term.
dg/dtAt, where At
=
We can express Ag as
T, the time step. We can then write that
A F --
kd T.
d
k(g) ag dt"
(10.40)
The Laplace Transform of dg/dt is sAg. Substituting this for dg/dt and taking the
limit as Ag approaches zero, we derive the stiffness due to sampling delay, denoted
krs, as
krs
OF
Fo Ok
Og
k(g)Og
Ts.
(10.41)
We have replaced Fd with FO for consistency with the other stiffness results.
We
therefnre see that. if the force-Hux relationship depends on gap, we will have a resulting
stiffness that increases with frequency even if our force feedforward is perfect and
our flux loop has infinite bandwidth. For our prototype actuator, if we use the k(g)
function described by (10.6), we derive a maximum k,, equal to 6.4 x 10-6s in units of
N/iim. This is computed at the maximum aituator force FO = 400 N and a sampling
rate of 20 kHz. This is well below the maximum allowable stiffness.
10.6.3
Force Pulse Experiments
To demonstrate the force accuracy of the reluctance actuator with flux control and
force-flux inversion, we attempted to have the reluctance actuator follow a desired
force pulse waveform. This force pulse waveform is a scaled version of the force wave-
406
form used for simulation in Chapter 4. It is therefore similar to the force feedforward
that would be used on a scanning lithography machine. This force waveform was set
as Fd in the block diagram of Figure 10-1.
The same force was fedforward to the
voice coil actuator to aid in maintaining the desired air gap. The nominal air gap
during the experiment was 530 pm. The hybrid flux estimation scheme included the
SHM observer. The reluctance actuator force was measured with the load cells and
then compared to the reference force.
Initial Force Pulse Results
Figure 10-39 shows the initial results of the force pulse experiment.
We see that
the maximum error between the reference force and measured force is 11.8 N. This
is 3.7% of full-scale force. The largest errors occur during the portions of the force
profile with high dF/dt. One unexpected result was the overshoot that occurs at the
end of the positive dF/dt portion of the force pulse. This overshoot is followed by a
slow decay to the correct force value. The time constant associated with this decay
is much slower than the time constant associated with the flux loop bandwidth.
We hypothesized that the cause of this overshoot and slow decay is the loopwidening phenomenon associated with the sense coil measurement documented in
Chapter 9.
Because of the phase lag that was discovered in the sense coil mea-
surement, the flux controller will drive the actuator harder (i.e., larger currents) to
compensate for the measured flux lag. This lag will be the most pronounced during
the high dF/dt portions of the force pulse because this is the portion of the force
pulse with the highest frequency content. Because the measured force was not found
to exhibit this same phase lag, the force will feature an overshoot. After the high
dF/dt portion of the pulse has completed, the flux controller transitions to using the
hysteresis model to estimate the flux rather than the sense coil measurement. This
is a result of the complementary filter structure. Because the hysteresis model does
not include any loop-widening phenomenon, the flux controller reduces the actuator
current until the hysteresis model output matches the desired flux. The measured
force is likewise reduced.
407
Force
40011
1
1
-
300-
reference force
measured farce
-
a 200
100
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.25
0.3
0.35
0.4
Force Error
5-
0
-
<l-5
-10 --
0
0.05
0.1
0.15
0.2
time [sec]
Figure 10-39: Force pulse experimental results with flux feedback and force-flux inversion. TOP: reference force profile and measured force profile. BOTTOM: error
between reference force and measured force.
Figure 10-40 shows the position error during the force pulse experiment. The air
gap changes by over 10 pm during the experiment. The voice coil feedforward assumes
that the air bearing slide is a pure mass. However, because of the high frequency
content of the reluctance actuator force pulse, this is no longer a good approximation
and the dynamics of the air bearing slide affect the results. The position error could
be reduced by including feedforward that takes into account the plant dynamics.
Another possibility would be to use iterative learning control to improve the voice
coil feedforward signal over multiple trials.
Force Pulse Results with Increased Complementary Filter Pair Cutoff Frequency
To test the hypothesis that the overshoot and slow decay in the force measurement
were the result of the sense coil loop widening, we changed the cutoff frequency for the
complementary filter pair from 1 Hz to 10 Hz. This will cause the hybrid flux estimator
to rely mainly on the hysteresis model instead of the sense coil measurement over a
408
0.545
0.540.535E
0.53
Ca
0.5250.520.515
0
0.1
0.2
0.3
0.4
time [sec]
Figure 10-40: Air gap measurement during force pulse experiment.
larger frequency range (up to 10 Hz instead of 1 Hz). Doing this should reduce the
dependency of the flux estimate on the sense coil during the high dF/dt portion of
the force pulse, and so we expect to see a reduced overshoot. Figure 10-41 shows the
results of this experiment. The overshoot decreases from the previous results. The
maximum error is now 8.6 N, or 2.7% of full-scale force. Moreover, the time constant
associated with the overshoot decay is faster. These results provide confirmation that
the loop widening is the cause of the force overshoot.
One potential solution to the overshoot problem is to increase the complementary filter pair cutoff frequency until the flux estimator's reliance on the sense coil is
sufficiently reduced such that the overshoot reaches acceptable levels. However, this
defeats the very purpose of having the sense coil in the first place. Moreover, with
the SHM observer having a crossover frequency of 2 kHz, the accuracy of the SHM
estimate will suffer at higher frequencies. Another alternative solution is to calibrate
the phase lag between the sense coil measurement and force measurement and then
incorporate this dynamic mapping into the force inversion.
A third alternative is
to use iterative learning control to find the correct signal with which to drive the
reluctance actuator to achieve the desired force. One challenge to this is that ILC
requires a repeatable signal, but the gap disturbance will not necessarily be repeatable. Thus, the signal required to drive the reluctance actuator will change between
409
Force
4 00
3 00
-- reference frc
-es
-
2 0000-
00
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.25
0.3
0.35
0.4
Force Error
.1 10
5 --
0
-5-
00
0.05
0.1
0.15
0.2
time [sec]
Figure 10-41: Force pulse experimental results with complementary filter pair cutoff frequency set to 10 Hz. TOP: reference force profile and measured force profile.
BOTTOM: error between reference force and measured force.
trials. However, since we attained acceptable levels for the flux-controlled reluctance
actuator stiffness using the sense coil, it is possible that the flux controller will be
able to compensate for the gap disturbance in spite of the loop-widening problem.
The best solution is of course to solve the loop-widening problem so that there is no
phase lag between the flux measured by the sense coil and the measured force.
Force Pulse Results with Applied Gap Disturbance
We tested the compensated reluctance actuator's ability to track a force pulse in the
presence of a gap disturbance similar to a gap disturbance on a scanning lithography
stage. The measured gap disturbance is shown in Figure 10-42. The peak-to-peak
amplitude of the disturbance has a maximum of about 30 pm.
The measured force pulse and the associated force error are shown in Figure 1043. The complementary filter pair cutoff frequency was 10 Hz. The error looks very
similar to the error without the applied gap disturbance from Figure 10-41, indicating
that the flux loop and force inversion are able to reject the gap disturbance. However,
410
0.55
0.5450.54
E0.535
E
CL
cu
0.530.525
0.52
0.515
0
0.1
0.2
0.3
0.4
time [sec]
Figure 10-42: Gap disturbance during force pulse experiment.
because there is still a sizable position error even when not intentionally applying a
gap disturbance (see Figure 10-40), we are not truly comparing the force accuracy with
and without gap disturbances. Nevertheless, the fact that there is not much change
in the force error is a promising result. Another difficulty with this measurement is
that we are limited by the load cell charge amplifier: since we are measuring large
dynamic forces in the force pulse test, the charge amplifier sensitivity must be lowered
to avoid overloading. This limits the minimum force change we are able to detect.
10.7
Summary
In this chapter, we presented experimental results of our reluctance actuator prototype on the 1-DoF air bearing testbed. We tested the SHM observer as well as flux
feedback loop that incorporated it. For the flux feedback loop, we designed a complementary filter pair that permitted us to take advantage of the benefits of both a sense
coil measurement and the hysteresis model estimate. One subtlety we discovered is
that the complementary filter pair must be at least second-order to avoid offsets in the
sense coil measurement translating into flux estimation errors. We tested the force
accuracy of the reluctance actuator using the flux feedback loop and a force-flux-gap
model inversion implemented as a lookup table.
411
Force
40
-
- ---
30 0-
reference force
measured force
820 010 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.25
0.3
0.35
0.4
Force Error
10
-
5
L
0
0
0.05
0.1
0.15
0.2
time [sec]
Figure 10-43: Force pulse experimental results with gap disturbance applied. TOP:
reference force profile and measured force profile. BOTTOM: error between reference
force and measured force.
We demonstrated that the SHM observer worked in real-time operation and could
accurately estimate the actuator flux. We calibrated the SHM at a gap of 530 Pm.
We demonstrated that the SHM was capable of estimating the flux when we offset
the nominal gap by 50 pm to 580 lpm. The SHM was also shown accurately to predict
the flux in the face of a dynamically varying gap with a peak-to-peak amplitude of
35 vm and was demonstrated to be numerically stable in real-time in the face of such
a gap disturbance.
We experimentally demonstrated a flux control loop crossover frequency of 4 kHz
that was capable of reducing the stiffness of the reluctance actuator to well below
the maximum allowable stiffness. For example, at 10 Hz, the flux-controlled actuator
exhibited a stiffness of 0.003 N/pm at a bias force of 35 N, which is an order of
magnitude below the maximum stiffness specification of 0.04 N/Pm. We also showed
that using the SHM observer in our hybrid estimation scheme results in lower stiffness
than using the Chua model.
The dynamic force tracking capability of the flux loop was demonstrated via a
412
measured frequency response and via force pulse experiments in the time domain.
The force tracking capability did not meet the required 99.9% accuracy requirement.
We determined that one of the likely reasons for this is the loop-widening phenomenon
exhibited by the sense coil measurement. Resolving this problem would result in more
accurate force tracking. We learned that one way to improve the force tracking in
the face of the loop widening behavior was to increase the complementary filter pair
cutoff frequency so that the flux estimator would rely more on the hysteresis model
estimate than on the sense coil measurement.
In the next chapter, we will present our thesis conclusions and recommendations
for future work. Included in our recommendations are suggestions for how to improve
the force tracking capabilities of the reluctance actuator.
413
414
Chapter 11
Conclusion and Future Work
11.1
Conclusion
This thesis develops control and modeling techniques for a reluctance actuator in
order to achieve high-dynamic-force accuracy on a scanning photolithography stage.
The primary contributions of this thesis include:
1. Designed and experimentally demonstrated a flux controller for a reluctance
actuator that utilizes a sense coil for the feedback measurement.
2. Experimentally investigated the high-force-accuracy capability of a reluctance
actuator using flux control.
3. Experimentally demonstrated the low stiffness of a reluctance actuator using
flux control.
4. Designed and experimentally demonstrated a low-frequency flux hysteresis estimate based on the actuator current and air gap to compensate for the poor
performance of the sense coil at low frequencies. This low-frequency estimate
was combined with the sense coil estimate using a hybrid estimation scheme.
5. Developed a novel way to model actuator hysteresis, the sheared hysteresis
model (SHM), that takes advantage of the linearizing effect of the air gap.
415
6. Developed an actuator hysteresis model that can incorporate the effect of eddy
currents.
7. Designed an adaptive gain observer that accurately models actuator hysteresis
over the entire B-H plane into saturation.
8. Investigated electromagnetic hysteresis dependence on frequency.
9. Developed simple approximate formulas for predicting reluctance actuator force
errors from hysteresis, eddy currents, and gap disturbances; for predicting actuator uncompensated stiffness and compensated stiffness; and for predicting
power dissipation from hysteresis and eddy currents.
11.2
Future Work
In this section, we outline suggestions for future work.
11.2.1
Further Investigation into Force Accuracy
Because of the sense coil loop-widening phenomenon, we were unable to achieve 99.9%
fnroe accuracy with the prototype rel
anc
ctuao"r. For aasqnse coil to be a viable
feedback sensor for the reluctance actuator, this problem must be better understood
and resolved.
Other ways to improve the force accuracy were not investigated owing to time
constraints. ILC could be used to find the optimal force command that achieves the
best force accuracy for the desired profile. ILC could potentially also be used to find
the hysteresis model parameters in an automatic way.
Flux feedforward could be added to the flux feedback loop to improve the flux
estimation accuracy, and thus, the force accuracy.
This feedforward could be as
simple as adding resistance compensation, whereby the voltage required to drive the
actuator resistance is fedforward to the actuator, or as complex as a full inverse
hysteresis model. One possibility for an inverse hysteresis model is to use the observer
416
structure developed in this thesis, but modifying it so that the reference signal is
the desired flux density rather than the measured current and placing the hysteresis
model in the observer feedback path. An advantage of this is that an inverse hysteresis
model does not have to be directly modeled. An alternative possibility is to use an
inverse hysteresis model that is simpler than the one used for the low-frequency flux
estimation for the flux feedback. Since the high-bandwidth flux feedback loop should
be designed such that it will not be the limiting factor in achieving the force accuracy
and stiffness specifications, any flux feedforward does not need to be highly accurate,
as its function is simply to give incremental improvements.
11.2.2
Analog or FPGA Flux Control Loop
On our testbed, we were limited in the flux loop bandwidth we could achieve by the
sampling rate of the dSPACE RTC. Our accuracy was also limited by the resolution
and noise of the ADC. Designing an analog control loop or a controller implemented
in an FPGA could easily permit loop bandwidths into the tens of kHz. Designing
an amplifier tailored to driving the reluctance actuator rather than a commercial
amplifier could also increase bandwidth and accuracy. A hybrid scheme could also be
pursued: the low-frequency hysteresis model estimate could be computed on a slow
RTC while the controller, sense coil integration, and complementary filter pair are
implemented in the analog domain or on a high-speed FPGA.
11.2.3
Optimized Actuator Design
For incorporation into an actual lithography stage, it is important to optimize the
actuator for high force-to-mass ratio in order to achieve the required accelerations
and servo bandwidths. One straightforward way toward reducing the actuator mass
is to use CoFe as the core and target material rather than NiFe, because CoFe has a
much higher saturation flux density. Incorporating an efficient water-cooling jacket
would allow the actuator to run much higher current densities and would permit a
much more compact design. Using a double-sided actuator, described in Section 2.6.1,
417
could allow for half the total target mass on the moving stage since each target now
only needs to transfer half the force compared to the equivalent single-sided design.
Geometric optimization also should be considered. We briefly remarked on a couple
possibilities for geometric optimization in Section 8.4.6. Other possibilities in addition
to these should also be investigated.
11.2.4
Cross-Coupling
With our testbed we were only capable of characterizing the actuator in the driving
direction. Investigation into the other degrees-of-freedom should also be analyzed and
tested. These include parasitic torques and forces and cross-axis stiffnesses. In [62],
which supplements the work of this thesis, the parasitic torques resulting from angle
disturbances have been investigated.
11.2.5
Integration into 6-DoF Stage
To prove the reluctance actuator fully for scanning lithography applications, the actuator must be integrated into a full magnetically-levitated 6-DoF stage and used for
position control for scanning profiles. Key issues that need to be addressed include
calibration on the 6-DoF stage, achieving small operating gaps given assembly and
crash tolerances, determining whether a 'pull-only' design is acceptable or whether a
'push-pull' design is needed, and managing the additional sensor wires when mounted
on a high-acceleration stage.
11.2.6
Other Applications
We briefly mention how a couple of the concepts presented in this thesis might be
applied to other applications.
Hybrid Estimation Scheme
The hybrid flux estimation scheme detailed in Chapter 10 could be applied to other
applications in which it would be advantageous to combine the benefits of a sensor or
418
estimation method suited to low frequencies with the benefits of a sensor or estimation
method suited to high frequencies. In [4], such a scheme is applied to a piezoelectric
actuator, combining a hysteresis model with charge control. On the reluctance actuator, we could combine a Hall sensor with the sense coil using this scheme. If force
sensing were considered instead of flux sensing, the AC-coupled quartz piezo load
cells could be combined with a force-current hysteresis model based on the current
and gap measurements. We can also think of applications such as those involving a
rotating mechanical system that includes a transmission between the drive motor and
the load. Suppose the output is measured by a tachometer, but we want to control
the position. We can use a hybrid estimation scheme to combine the integral of the
tachometer signal with an estimate of the position based on the input signal and a
model of the transmission.
Observer Structure
We used an observer to estimate the flux of the actuator in the presence of a gap
disturbance. One of the benefits of this method is that by treating the gap variation
as a disturbance to the observer loop, we were able to estimate the flux without having
to resort to a more complex two-input hysteresis model. This idea can be extended
to other nonlinear phenomena with more than one input.
Remaining within the
domain of reluctance actuators, the flux dependency on temperature could possibly
be modeled in this way.
For piezoelectric actuators, if both the electric field and
applied stress levels are changing, the voltage-strain relationship could be modeled
with a hysteresis model, and the stress variation could be handled as a disturbance
to an observer loop. Similarly, the dependence of magnetostriction on both magnetic
field and stress could possibly be modeled in this way. Shape memory alloy's dual
dependence on temperature and stress could also potentially be modeled in this way.
419
420
Bibliography
[1] HJMTS Adriaens, W De Koning, and Reinder Banning. Modeling piezoelectric
actuators. Mechatronics, IEEE/ASME Transactions on, 5(4):331-341, 2000.
[2] Hyo-Sung Ahn, YangQuan Chen, and Kevin L Moore. Iterative learning control:
brief survey and categorization. IEEE TRANSACTIONS ON SYSTEMS MAN
AND CYBERNETICS PART C APPLICATIONS AND REVIEWS, 37(6):1099,
2007.
[3] Farid Al-Bender, Vincent Lampaert, and Jan Swevers. The generalized maxwellslip model: a novel model for friction simulation and compensation. Automatic
Control, IEEE Transactions on, 50(11):1883-1887, 2005.
[4] Darya Amin-Shahidi. Design and control of a self-sensing piezoelectric reticle
assist device. PhD thesis, Mechanical Engineering, Massachusetts Institute of
Technology, 2013.
[5] Darya Amin-Shahidi, Ian MacKenzie, and David L Trumper. Magnetic flux
linkage estimation and control for a reluctance actuator. In Proc. of 2013 ASPE
Spring Topical Meeting.
[6] Analog Devices. http: //
. analog.com/en/index.
html.
[7] M Angeli, E Cardelli, and E Della Torre. Modelling of magnetic cores for power
electronics applications. Physica B: Condensed Matter, 275(1):154-158, 2000.
[8] B Azzerboni, E Cardelli, E Della Torre, and G Finocchio. Reversible magnetization and lorentzian function approximation. Journal of applied physics,
93(10):6635-6637, 2003.
[9] B Azzerboni, E Cardelli, G Finocchio, and F La Foresta. Remarks about Preisach
function approximation using lorentzian function and its identification for nonoriented steels. IEEE transactions on magnetics, 39(5):3028-3030, 2003.
[10] BEI Kimco Magnetics. www. beikimco. com.
[11] Y Bernard, E Mendes, and F Bouillault. Dynamic hysteresis modeling based on
Preisach model. Magnetics, IEEE Transactions on, 38(2):885-888, 2002.
421
[12] Y Bernard, E Mendes, and Z Ren. Determination of the distribution function
of Preisach's model using centred cycles. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering,
19(4):997-1006, 2000.
[13] Giorgio Bertotti. General properties of power losses in soft ferromagnetic materials. Magnetics, IEEE Transactions on, 24(1):621-630, 1988.
[14] Giorgio Bertotti. Hysteresis in magnetism: for physicists, materials scientists,
and engineers. Gulf Professional Publishing, 1998.
[15] Richard Boll. Soft Magnetic Materials: fundamentals, alloys, properties, products, applications. Heyden, 1979.
[16] Richard Boll, editor. Soft magnetic materials: fundamentals, alloys, properties,
products, applications/ The Vacuumschmelze Handbook. Heyden, London, 1979.
[17] Douglas A Bristow, Marina Tharayil, and Andrew G Alleyne. A survey of iterative learning control. Control Systems, IEEE, 26(3):96-114, 2006.
[18] Hans Butler. Position control in lithographic equipment. IEEE Control Systems
Magazine, 31(5):28-47, 2011.
[19] Hans Butler and Franciscus Adrianus Gerardus Klaassen. Method for controlling the position of a movable object, a positioning system, and a lithographic
apparatus, 2008. US Patent 7,352,149.
[20] E Cardelli, F Bertoncini, S Di Fraia, and B Tellini. Implementation of the
modified Preisach scalar model in the finite difference-time-domain numerical
modeling. Physica B: Condensed Matter, 306(l):126-131, 2001.
[21] Ermanno Cardelli, Edward Della Torre, and Enrico Pinzaglia. Identifying the
preisach function for soft magnetic materials. IEEE transactionson magnetics,
39(3):1341-1344, 2003.
[22] Ping-Wei Chang, Kazuhiro Hirano, Tim Teng, Pai-Hsueh Yang, and Bausan
Yuan. E/i core actuator commutation formula and control method, August 7
2007. US Patent 7,253,576.
[23] L Chua and Steven C Bass. A generalized hysteresis model.
IEEE Transactionson, 19(1):36-48, 1972.
[24] Leon 0 Chua, Charles A Desoer, and Ernest S Kuh.
circuits. McGraw-Hill New York, 1987.
Circuit Theory,
Linear and nonlinear
[25] Leon 0 Chua and Keith A Stromsmoe. Lumped-circuit models for nonlinear
inductors exhibiting hysteresis loops. Circuit Theory, IEEE Transactions on,
17(4):564-574, 1970.
422
[26] Leon 0 Chua and Keith A Stromsmoe. Mathematical model for dynamic hysteresis loops. InternationalJournal of Engineering Science, 9(5):435-450, 1971.
[27] Bernard D Coleman and Marion L Hodgdon. A constitutive relation for rateindependent hysteresis in ferromagnetically soft materials. Internationaljournal
of engineering science, 24(6):897-919, 1986.
[28] COMSOL Multiphysics Modeling Software. http: //www . comsol . com/.
[29] Crown International, Inc. http: //www. crownaudio. com/.
[30] Crown International, Inc. Instruction Manual: DC-300A Dual Channel Laboratory Amplifier.
[31] Philip R Dahl. Solid friction damping of mechanical vibrations. AIAA Journal,
14(12):1675-1682, 1976.
[32] C Canudas De Wit, Hans Olsson, Karl Johan Astr6m, and Pablo Lischinsky.
A new model for control of systems with friction. Automatic Control, IEEE
Transactions on, 40(3):419-425, 1995.
[33] Edward Della Torre. Magnetic hysteresis. IEEE press New York, 1999.
[34] B.G. Dijkstra. Iterative learning control, with applicationsto a wafer stage. PhD
dissertation, Delft University of Technology, Mechanical Engineering Systems
and Control Group, September 2004.
[35] dSPACE Inc. www.dspace.com/en/inc/home.cfm.
[36] DuPont. http: //www. dupont. com/.
[37] Michel Dussaux. The industrial applications of the active magnetic bearings
technology. In Proc. 2 nd Int. Symp. Magn. Bearings, pages 33-38, 1990.
[38] EpoxySet, Inc. http://www.epoxysetinc.com/.
[39] Hermann A Haus and James R Melcher.
Electromagnetic fields and energy.
Prentice Hall Englewood Cliffs, 1989.
[40] Kazuhiro Hirano, Tim Teng, Pai Hsueh Yang, and Bausan Yuan.
Autocalibration of attraction-only type actuator commutations, April 25 2006. US
Patent 7,034,474.
[41] Sven Antoin Johan Hol, Antonius Franciscus Johannes De Groot, Theodorus
Petrus Maria Cadee, Marijn Kessels, and Daniel Godfried Emma Hobbelen. Actuator, positioning system and lithographic apparatus, June 25 2013. US Patent
8,472,010.
423
[42] Sven Antoin Johan Hol, Jan Van Eijk, Johannes Petrus Martinus Bernardus
Vermeulen, Gerard Johannes Pieter Nijsse, Simon Bernardus Cornelis Maria
Martens, Yang-Shan Huang, Michael Wilhelmus Theodorus Koot, Jeroen
De Boeij, Maarten Hartger Kimman, Wei Zhou, et al. Electromagnetic actuator,
stage apparatus and lithographic apparatus, April 1 2014. US Patent 8,687,171.
[43] Yi-Ping Hsin.
Adaptive gain adjustment for electromagnetic devices, May 16
2006. US Patent 7,046,496.
[44] SYR Hui and J Zhu. Numerical modelling and simulation of hysteresis effects
in magnetic cores using transmission-line modelling and the Preisach theory. In
Electric Power Applications, IEE Proceedings-, volume 142, pages 57-62. IET,
1995.
[45] Faygal Ikhouane, Victor Mafiosa, and Jose Rodellar. Dynamic properties of the
hysteretic bouc-wen model. Systems & control letters, 56(3):197-205, 2007.
[46] Mohammad Imani Nejad. Self-bearing motor design & control. PhD thesis,
Mechanical Engineering, Massachusetts Institute of Technology, 2013.
[47] DC Jiles and DL Atherton. Theory of ferromagnetic hysteresis.
magnetism and magnetic materials, 61(1):48-60, 1986.
[48] John G Kassakian, Martin F Schlecht, and George C Verghese.
power electronics. Addison-Wesley Reading, USA, 1991.
Journal of
Principles of
[49] Andelko Katalenid. Control of reluctance actuatorsfor high-precisionpositioning.
PhD dissertation, Eindhoven University of Technology, 2013.
~F[5tjU(1
TJ_
T__ -
1ejpc, 11c. 11L
p:- / / www .Kepcopower. corn,.
com/.
[51] Byeongil Kim and Gregory N Washington. Nonlinear position control of smart
actuators using model predictive sliding mode control. In A SME 2008 Conference
on Smart Materials, Adaptive Structures and Intelligent Systems, pages 511-522.
American Society of Mechanical Engineers, 2008.
[52] Kistler. www.kistler.com.
[53] Daniel Joseph Kluk. An advanced fast steering mirror for optical communication.
Master's thesis, Massachusetts Institute of Technology, 2007.
[54] Stephan Kotyczka. Manufacturing and control of a single degree of freedom
stage. Visiting student at MIT.
[55] Klaus Kuhnen and Hartmut Janocha. Adaptive inverse control of piezoelectric
actuators with hysteresis operators. In Proceedings of European Control Confer-
ence (ECC), 1999.
424
[56] Xiaodong Lu. Electromagnetically-drivenultra-fast tool servos for diamond turning. PhD thesis, Mechanical Engineering, Massachusetts Institute of Technology,
2005.
[57] Magnetic Metals. Tape wound core design manual. www. magmet . com.
Transformer
[58] Magnetic Metals.
lamination/materials .php.
lamination
catalog.
www
.magmet . com/
[59] H Malsky. Printed winding DC torque motor. Technical report, Charles Stark
Draper Laboratory, Inc., 1976.
[60] MathWorks. www.mathworks.com/products/matlab/.
[61] Isaak D Mayergoyz. Mathematical models of hysteresis and their applications.
Academic Press, 2003.
[62] Roberto Mel6ndez. A reluctance actuator gap disturbance testbed.
thesis, Massachusetts Institute of Technology, 2014.
Master's
[63] Kevin L Moore. Iterative learning control for deterministic systems. Springer,
1993.
[64] National Instruments. http: //www. ni. com/labview/.
[65] New Way Air Bearings. www. newayairbearings. com.
[66] Donald Richard Nohavec. Magnetic bearing design for interferometric mirrorscanning mechanisms. Master's thesis, Massachusetts Institute of Technology,
1997.
[67] Kazuya Ono. Curved i-core, June 14 2005. US Patent 6,906,334.
[68] Vlado Ostovic.
Dynamics of saturated electric machines, volume 2. Springer-
Verlag New York, 1989.
[69] Pacific Scientific-OECO. http: //f wbell. com/def ault . aspx.
[70] David J Perreault and Khurram Afridi. 6.332 Advanced Topics in Power Elec-
tronics. Fall 2013. MIT.
[71] Tony Poovey. A kinematically coupled magnetic bearing test fixture. Master's
thesis, University of North Carolina at Charlotte, 1992.
[72] James K Roberge. Operational amplifiers: Theory and practice john wiley and
sons. Inc., New York, London, Sydney, Toronto, 1975.
[73] Iuliana Rotariu, Branko Dijkstra, and Maarten Steinbuch. Standard and lifted
approaches of iterative learning control applied on a motion system. In Proceedings of The International Symposium on the Mathematical Theory of Networks
and Systems (MTNS 2004), 2004.
425
[74] HC Roters. Electromagnetic devices, 1941.
[75] Stephen Roux, Frederick Michael Carter, and Ross Ian Mackenzie. Method for
controlling the position of a movable object, a control system for controlling a
positioning device, and a lithographic apparatus, October 8 2013. US Patent
8,553,205.
[76] Ralph C Smith. Smart material systems: model development, volume 32. Siam,
2005.
[77] Mohammad Sheikh Sofia, Seyed Mehdi Rezaei, and Mohammad Zareinejad.
Adaptive trajectory tracking control of nanopositioning stage under dynamic
load condition. In ASME 2009 InternationalMechanical Engineering Congress
and Exposition, pages 315-320. American Society of Mechanical Engineers, 2009.
[78] Sony Precision Technology, Inc. sony-magnescale. com.
[79] Gilbert Strang and Kaija Aarikka. Introduction to applied mathematics, volume 16. Wellesley-Cambridge Press Wellesley, MA, 1986.
[80] Tektronix, Inc. www .t ek. c om.
[81] Tektronix, Inc. AM502 Differential Amplifier Instruction Manual, 1986.
[82] Ting-Chien Teng and Bausan Yuan. Magnetic actuator producing large acceleration on fine stage and low rms power gain, October 10 2000. US Patent
6,130,517.
[83] Ting-Chien Teng and Bausan Yuan. Stage having paired e/i core actuator con-
trol, May 30 2000. US Patent 6,069,417.
[84] David L Trumper. Nonlinear compensation techniques for magnetic suspension
systems. In NASA Workshop on Aerospace Applications of Magnetic Suspension
Technology, 1990.
[85] David L Trumper, Sean M Olson, and Pradeep K Subrahmanyan. Linearizing control of magnetic suspension systems. Control Systems Technology, IEEE
Transactionson, 5(4):427-438, 1997.
[86] Ferenc Vajda and Edward Della Torre.
of complete-moving-hysteresis models.
Efficient numerical implementation
Magnetics, IEEE Transactions on,
29(2):1532-1537, 1993.
[87] Jozef V6r6s. Modeling and identification of hysteresis using special forms of the
coleman-hodgdon model. Journal of Electric Engineering, 60(2):100-105, 2009.
[88] Wen-Fang Xie, Jun Fu, Han Yao, and C-Y Su. Neural network-based adaptive
control of piezoelectric actuators with unknown hysteresis. InternationalJournal
of Adaptive Control and Signal Processing, 23(1):30-54, 2009.
426
[89] Jian-Xin Xu, Sanjib K Panda, and Tong Heng Lee. Real-time iterative learning
control: design and applications. Springer, 2008.
[90] Markus Zahn. 6.641 Electromagnetic Fields, Forces, and Motion. Spring 2008.
MIT.
[91] Markus Zahn. Electromagneticfield theory: a problem solving approach. Krieger
Pub. Co., Malabar, 2003.
427